Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations

This is a collection of open problems concerning various areas in function theory, functional analysis, theory of linear and nonlinear partial differential equations.


Introduction
The title of this article was inspired by the title of Sholem Aleichem's story "Seventy five thousand". Alas, the reader will find only seventy five problems here! I hope each of them can produce a long sequence (a thousand) of others. The problems below present a sample of questions that I came across upon during many years of research, but have never seen them solved or even clearly formulated. Some of them might turn out being rather simple, while others would present a significant challenge and hopefully attract interest of beginning researchers and experts alike.
I tried to select questions that can be formulated without much preliminary knowledge. For more explanations the reader is directed to the cited literature.
The problems included here address both linear and non-linear PDEs. One can hear sometimes that "all linear problems had been already solved". Hopefully, the following list of problems will show that this opinion is mistaken.

Basic Notations
Here some notations which are used in the article are collected.
By a domain Ω we mean an open connected set in R n with boundary ∂Ω. We say that Ω is a Lipschitz graph domain if it can be locally explicitly represented in a certain cartesian system by a Lipschitz function. The domain is weakly Lipschitz if it is locally Lipschitz diffeomorphic to a ball. Let H n−1 stand for the (n − 1)-dimensional Hausdorff measure. Let B r (x) denote the open ball with center x and radius r, B r = B r (O), where O is the origin. We shall use the notation m n for n-dimensional Lebesgue measure and v n for m n (B 1 ). We use the notation ∇ l u for the distributional gradient of order l, i.e. ∇ l u = {D α u}, where α is a multi-index (α 1 , . . . , α n ), D α = D α1 x1 . . . D αn xn and |α| = j α j = l. We put ∇ = ∇ 1 .
Let 1 ≤ p < ∞, l ∈ N, and let Ω be a domain in R n . By L l,p (Ω) we denote the space of distributions whose gradient of order l is in L p (Ω) supplied with the norm where ω is a bounded subdomain of Ω with dist(∂ω, ∂Ω) > 0.
We call the quantities a and b equivalent if there are positive constants c 1 and c 2 such that By c, C we denote different positive constants. The Einstein summation convention is always assumed.

Isoperimetric Problem for Fractional Perimeter
Let Ω be a bounded domain with smooth boundary ∂Ω and α ∈ (0, 1). The following set function was considered in [68]: Later, P α (g) was called the "fractional perimeter of Ω" and has attracted a lot of attention since.
By [68,Section 6], (see also [71,Section 11.10]), the best constant C in the isoperimetric inequality is the same as the best constant C in the functional inequality where q = n/(n − α) and u ∈ C ∞ 0 (R n ). By [68, p. 333], The constant factor on the right, although explicit, is hardly best possible. Problem 1. Find the best value of the isoperimetric constant C in (3.1).

Strong Capacitary Inequality for Functions with Arbitrary Boundary Data
Let F be a relatively closed subset of Ω. By the capacity of F generated by the norm in L l,p (Ω) we mean the set function Similar capacities are frequently used in potential theory, partial differential equations and theory of function spaces (see [60,69,70]). Here is, for example, a simple application of capacity to the integral inequality 3) where p ∈ (1, ∞), μ is a Borel measure on Ω and u ∈ C ∞ (Ω) is an arbitrary function. Take a relatively closed subset F of Ω and any function u from the above definition of cap(F, L l,p (Ω)). Then (3.2) implies Thus, this isocapacitary isoperimetric inequality proves to be a necessary condition for inequality (3.3).
It is easy to show that the condition (3.4) is sufficient for (3.3) provided that the so-called strong capacitary inequality For functions with zero boundary conditions inequalities of type (3.5) first appeared in [61] (see also [69] and [1]). If l = 1, (3.5) holds without restrictions on Ω. However, already for l > 1 the domain should satisfy some regularity conditions as the counterexample constructed in [80] shows (see also [83,Section 2.13]). This gives rise to the following Problem 2. Find necessary and sufficient conditions, or only non-trivial sufficient conditions, on Ω guaranteeing the inequality (3.5).

Boundary Traces of Functions in a Sobolev Space
Let Ω be a bounded weakly Lipschitz domain and let ϕ : It is well-known that ϕ is the trace of a function in L 1,p (Ω), if and only if ϕ ∈ L 1 (∂Ω) for p = 1 and ϕ ∈ L 1−1/p,p (∂Ω) for p > 1, where the Besov space L α,p (∂Ω), 0 < α < 1, is supplied with the seminorm This fact fails if Ω is not weakly Lipschitz. Other characterizations of the trace are given for some classes of non-Lipschitz domains in [83,Chapter 6]. L p (Ω) such that for "quasi-every" pair of points x and x on ∂Ω and for any locally rectifiable arc γ ⊂ Ω joining x and x there holds the inequality By "quasi-everywhere" one means outside of a set of zero L 1,p -capacity.

Embedding of a Sobolev Space into the Space of Distributions
Let Ω be an open set in R n . By L l,p 0 (Ω, μ) we denote the completion of C ∞ 0 (Ω) in the norm The question of embedding L l,p 0 (Ω, 0) into the distribution space [C ∞ 0 (Ω)] is completely solved in [23] for p = 2 and by [69,Theorem 15.2] for all values of p.

An Approximation Problem in the Theory of Sobolev Spaces
Lewis [43] showed that C ∞ (Ω) is dense in L 1,p (Ω) if Ω is an arbitrary Jordan domain in the plane, i.e. ∂Ω is homeomorphic to ∂B 1 . In [44] he asked if C ∞ (Ω) is dense in L l,p (Ω) with l > 1 for every Jordan domain Ω ⊂ R 2 . It is almost obvious that L ∞ (Ω) ∩ L 1,p (Ω) is dense in L 1,p (Ω) for every open set Ω ⊂ R n . A direct extension of this property for L l,p (Ω) with l > 1 is impossible, as shown in [80]. A planar domain Ω in the counterexample proposed in [80] is not Jordan, which gives rise to the following Problem 5. Are bounded functions belonging to L l,p (Ω), l > 1, dense in L l,p (Ω) if Ω is an arbitrary Jordan domain?

Extensions of Sobolev Functions Outside of an Angle
Let A α denote the angle {z ∈ C : 0 < argz < α}, 0 < α < 2π, and let E α be the extension operator L 1,2 (A α ) → L 1,2 (R 2 ) which has the least norm. Using the Radon transform, one can show that

Problem 6.
Find the least norm of the extension operator

Best Constants in Some Classical Inequalities
Consider the inequality where B 1 is the open unit ball in R n , n > 2, dω is an element of the surface area on the sphere ∂B 1 , and u is an arbitrary C 0,1 (Lipschitz) function given on B 1 that vanishes on a closed subset F of B 1 . This inequality holds if and only if the harmonic (Wiener) capacity cap(F ) is positive, where the Wiener capacity is defined as follows The following assertion containing an explicit value of C in (3.6) was obtained in [51], see also [91]. Let where c n = 2(n − 1)nv n . (3.8) Various inequalities related to (3.7) have interesting applications to the theory of partial differential equations and were studied from different points of view in [71] and elsewhere. However, the question of sharp constants has never been addressed. Therefore, it seems reasonable to formulate the problem.

Problem 7.
Find the best value of c n in (3.7).

Hardy Inequality with Capacitary Distance to ∂Ω
Let Ω be a domain in R n , n > 2. As is well-known, the Hardy inequality where u ∈ C ∞ 0 (Ω) and d(x) is the distance from x to ∂Ω, does not hold for any domain and conditions for its validity are known (see [69, p. 733-735] for references).

Uncertainty Principle for Divergence-Free Fields
The inequality is the so-called uncertainty principle in quantum mechanics, and the constant 4 is sharp.
Problem 9. What is the best constant C > 0 in the inequality where u = (u 1 , . . . , u n ) is an arbitrary divergence-free vector field? Answer the same question concerning the inequality where p ∈ (1, ∞) and div u = 0 in R n .

Normalization of an Anisotropic Riesz Potential Space
Let 1 < p < ∞ and let l = (l 1 , . . . , l n ), where 0 < l i < 1. By L l,p (R n ) we denote the generalization of the classical Riesz potential space normed by where μ(D) is the pseudodifferential operator with symbol It is shown in [48] (see also [24]) that the space L l,2 (R n ) can be endowed with the norm D l,2 u L 2 (R n ) , where This norm is useful in applications such as the description of multipliers in L l,2 (R n ) and the theory of Dirichlet forms.

Problem 11. Find the best constant in the inequality
It is proved in [10] (see also [69, p. 229]) that when p = 2, The method used in the proof of this result cannot be applied to the case p = 2.

Trace Inequality with a Non-Positive Measure
Let Ω be a bounded smooth domain in R n and let σ be a real measure, not necessarily positive, even a distribution, supported on ∂Ω.
Problem 12. Let n > p > 1 and let u be any function in C ∞ (Ω). Find necessary and sufficient condition on σ for the validity of the inequality A similar problem for the inequality with σ supported on Ω was solved in [25] for p = 2 and in [26] for an arbitrary p > 1.

Domains for Which L l,p (Ω) ∩ L ∞ (Ω) is a Banach Algebra
Let Ω be a domain in R n . We may ask whether the space L l,p (Ω) ∩ L ∞ (Ω) is an algebra with respect to pointwise multiplication. Clearly, this is the case for l = 1. Since Stein's extension operator from a Lipschitz graph domain Ω is continuous as an operator the above question has the affirmative answer for the union of a finite number of Lipschitz graph domains. For example, Ω can be a bounded domain having the cone property. However, by a counterexample proposed in [80] (see also [69, p. 120-121]), it turns out that the space L l,p (Ω) ∩ L ∞ (Ω) is generally not an algebra. Remark. One can easily check that W 2,p (Ω) ∩ L ∞ (Ω) is an algebra with respect to the pointwise multiplication if and only if

A Multi-Dimensional Integral Equation Which
can be Reduced to One-Dimensional Let ϕ and ψ be functions defined on a multi-dimensional domain Ω.

Problem 14. Develop a theory of solvability of the integral equation
where f is a given function and λ is a complex parameter.
Note that (4.1) can be reduced to a certain one-dimensional integral equation by using the so-called coarea formula, see [69].

A Singular Integral Operator in L p (R n )
Consider the singular integral operator A with the symbol where φ is a smooth real-valued function on ∂B 1 , and λ is a large real parameter.

Problem 15. Prove or disprove the estimate
where 1 < p < ∞, c depends on n, p and the function φ.

Fredholm Radius of the Harmonic Double Layer Potential
Considering the logarithmic harmonic potentials of the single and double layer on curves of "bounded rotation", J. Radon has introduced the notion of the essential norm and the Fredholm radius of a bounded operator in 1919.
The essential norm L of a linear bounded operator L acting on a Banach space B is defined as where {K} is the set of all linear compact operators on B.
The Fredholm radius R(L) of the operator L is the radius of the largest circle on the complex λ-plane centered at λ = 0 inside which I + λL is a Fredholm operator.
The essential norm L is related to R(L) by the inequality  [20] that the Fredholm radius of the operator L, acting on a Banach space B equipped with the norm · 0 , satisfies where L is calculated with respect to a norm · in the set N ( · 0 ) of all norms in B, equivalent to · 0 .
Král [35] and Burago and Maz'ya [5] showed that for a n-dimensional bounded domain D with finite variation of the solid angle ω D (p, E) (p ∈ ∂D, E ⊂ R n ) the essential norm of the harmonic double layer potential in the Banach space C(∂D) is References to other results in the same area can be found in [36,Chapter 7].

Potential Theory for Surfaces with Cusps
A detailed theory of boundary integral equations for harmonic and elastic singe and double layer potentials on curves with exterior and interior cusps was constructed in [93].

Problem 17. Develop a theory of multidimensional boundary integral equations for domains with inner and outer cusps.
First results concerning the multidimensional case can be found in [84][85][86].

Regularity of Domains in the Theory of Boundary Integral Equations
In [90] (see also [88,Chapter 15]) a regularity theory of classical boundary integral equations involving harmonic, elastic and hydrodynamic potentials on nonsmooth surfaces is presented. A goal of this theory is to establish solvability in fractional Besov spaces.

Problem 18. Obtain results on the solvability of just mentioned equations in
Sobolev spaces L l,p (∂Ω) with integer l under minimal requirements on ∂Ω.

Integral Equations of Potential Theory Under Interaction of Nearly Touching Particles
The following difficulty arises when using the boundary element method to study hydrodynamic interactions among particles in suspensions and other similar phenomena when the particles are nearly touching. Integration of kernels of integral operators on elements near the closest contact points destroys the accuracy of the numerical procedure.

Problem 19.
Find asymptotic representation for densities of harmonic, elastic, and hydrodynamic potentials satisfying the boundary integral equations by using the distance between particles as a small parameter.

Non-Classical Maximal Operator
Let the maximal operator M ♦ be defined by where f is locally integrable in R n and the barred integral stands for the mean value. It is obviously dominated by the Hardy-Littlewood maximal operator M . A simple property of M ♦ is the sharp pointwise inequality for functions of one variable where f Br(x) stands for the mean value of f in the ball B r (x).

Fourier p-Capacity and Potentials
It is difficult to overestimate the role of various capacities of sets in analysis and partial differential equations. Let us introduce the set function where p ∈ [1, ∞], F is the Fourier transform and K is a compact set in R n . It is natural to call (4.2) the Fourier capacity of K. If p = 2, this capacity is equal to (m n K) 1/2 .

Problem 21. Find upper and lower geometrical estimates of the Fourier pcapacity. Develop a theory of Fourier p-potentials
where μ is a measure, in spirit of [73][74][75].

Integral Inequality for the Integrable Laplacian
It is shown in [89] that the inequality where a i,j are constants and u is an arbitrary complex-valued function in , holds if and only if a 1,1 + a 2,2 = 0. Other results of a similar nature are obtained in [70], where in particular it is proved that the inequality where α j are real constants, holds for all real-valued functions u ∈ C ∞ 0 (R n ) if and only if n j=1 α j = 0.

Problem 22. I conjecture that the inequality
where n > 1, Φ is a smooth function outside the origin and positive homoge-

Hölder Regularity of a Boundary Point with Respect to an Elliptic Operator of Second Order
Let Ω be a bounded domain in R n , n ≥ 2. We fix a non-isolated point O ∈ ∂Ω as the origin. Let us say that a function u defined on Ω is α-Hölder continuous at O with α > 0 if it has a limit u(O) as x → O and there exists α such that By u ϕ we mean a bounded solution to the Dirichlet problem where ϕ is a bounded Borel function on ∂Ω and is a uniformly elliptic operator with a measurable symmetric coefficient matrix A. Basic facts concerning solvability of this problem can be found in [45]. We introduce the L-harmonic measure H L (x, B), where x ∈ Ω and B is a Borel subset of ∂Ω (see e.g. [ A necessary and sufficient condition for the Hölder regularity of O is contained in the following assertion proved in [72].
The point O ∈ ∂Ω is Hölder regular with respect to L if and only if, for some positive constants λ and C, for all r > 0 and x ∈ Ω ∩ B r . Let n > 2 and let cap denote the Wiener capacity defined in Sect. 3.7. The following condition, sufficient for the Hölder regularity of O and independent of L, was found in [52,54,55]: Under additional restrictions on the domain, this condition proves to be necessary for the Hölder regularity, see [54,104].
Clearly, (5.3) fails or works simultaneously for all operators L. However, it is not necessary for the Hölder regularity in general, see the counterexample in [66, p. 509-510]. Although (5.2) formally depends on L, it is natural to formulate the Problem 23. Prove or disprove that the Hölder regularity of a point is independent of the operator L.

Uniqueness in the Cauchy Problem for the Laplace Equation
and its Dirichlet data has zero of infinite order at the origin. Moreover, let the normal derivative satisfy ∂u ∂t where h is a function defined on [0, ∞) positive on (0, ∞) and such that h(r) ↓ 0 as r ↓ 0. One can ask for which h it is true that (5.4) implies u = 0.
The following answer to this question was obtained in [76]. Suppose h satisfies certain regularity conditions (not to be specified here), then (5.4) implies u = 0 if and only if By Kelvin's transform, the criterion (5.5) can be stated for harmonic functions in a ball. However, the method of proof given in [76] does not provide an answer to the following question. In this case, it is only known that the uniqueness follows from the estimate with a certain positive c (see [39]) and with c = 2 + ε, ε > 0, (see [41]).

De Giorgi-Nash Theorem for Equations with Complex-Valued Coefficients
Let Ω be a domain in R n . By the classical De Giorgi-Nash theorem, an arbitrary weak solution of the uniformly elliptic equation with measurable bounded real-valued coefficients is locally Hölder continuous. If the coefficients a ij are complex-valued and for all x ∈ Ω and ξ ∈ R n , this assertion is not true for dimensions n > 4. See [95, p. 391-393] for examples of equations whose weak solutions are even unbounded near an interior point of a domain.

Problem 25.
Prove or disprove that De Giorgi-Nash property holds for dimensions n = 3 and n = 4.
The Hölder continuity of u in the case n = 2 follows from [94].

Fractal Domains Versus Lipschitz Domains
During the last fifty years considerable progress has been made in the study of elliptic boundary value problems on Lipschitz graph domains (see [27] for a survey of this development). In particular, classes of solvability and estimates of solutions received considerable attention in this area. A Lipschitz graph domain is locally given by where ϕ satisfies the Lipschitz condition. Replacing here Lipschitz condition by Hölder condition, we arrive at the definition of a Hölder graph domain.
With the existing rich theory of boundary value problems in Lipschitz graph domains, the absence of similar results in Hölder graph domains seems ubiquitous.
Problem 26. Address this issue.

Dirichlet Problem in a Planar Jordan Domain
Let u be a variational solution of the Dirichlet problem with zero boundary data for the equation in Ω, where Ω is an arbitrary bounded Jordan domain in R 2 (see Sect. 3.5), f ∈ C ∞ 0 (Ω) and L 2m is a strongly elliptic operator with real constant coefficients. for every small ε > 0.
This problem is unsolved even for the case of the Laplace operator which is related to Brennan's conjecture [7].

Hölder Regularity of Capacitary Potential of n-Simensional Cube
It can be easily proved by a barrier argument that the harmonic capacitary potential of the cube {x : 0 ≤ x i ≤ 1, i = 1, . . . , n} belongs to a certain Hölder class C α(n) , 0 < α(n) < 1, which gives rise to the following Problem 28. Describe the asymptotic behaviour of α(n) as n → ∞.

Non-Lipschitz Graph Domains Which are Weakly Lipschitz
Recall that open sets, locally Lipschitz diffeomorphic to a half-space, are called weak Lipschitz domains. One can easily construct cones with boundary smooth outside the vertex and polyhedra which are not Lipschitz graph domains (see [33, p. 4]).

Problem 29.
Do the known results on elliptic equations in Lipschitz graph domains (see [27] and references therein) extend to arbitrary weakly Lipschitz domains?
It seems, there are no counterexamples to this question and some results concerning non-Lipschitz graph cones suggest the affirmative answer to it, see [33,87].

Self-Adjointness of the Laplacian in a Weighted L 2 -Space
Let ρ ∈ L 1 loc (R n ), ρ > 0, and let In 1990 Eidus [14] proved that for n ≥ 3 a necessary condition for the operator

Peculiarities of Planar C 1 Domains and Extensions of the Dirichlet Laplacian
By L 1,2 0 (Ω) we denote the completion of C ∞ 0 (Ω) in the norm of L 1,2 (Ω). By Δ we denote the closure of the Laplacian Δ : L 2,2 (Ω) ∩ L 1,2 0 (Ω) → L 2 (Ω). LetΔ be the Friedrichs extension of Δ and let Δ * denote the adjoint of Δ. It is well known that Δ = Δ =Δ = Δ * , if Ω is sufficiently smooth. This is not true in general if Ω is not in the class C 2 .
The following result was obtained in [56,63].
Here are two surprising consequences of the above statement. (i) There exists a planar C 1 domain Ω such that the closure of Δ is selfadjoint, but the estimate does not hold for all u in the domain ofΔ, i.e. Δ =Δ but Δ = Δ. (ii) The closure of the operator (1+ε) ∂ 2 x1 +∂ 2 x2 , ε > 0, can be non-selfadjoint whilst for the same Ω, the generalized solution u ∈ L 1,2 0 (Ω) of the equation Δ u = f satisfies (5.10) and, consequently, Δ = Δ =Δ.

Example.
Let where C is a positive constant. Then Concerning the index of elliptic boundary value problems in domains with angle and cone vertices see [15,82].

Pointwise Estimates for Polyharmonic Green's Functions
Let Ω be an arbitrary domain in R n . Green's function of Δ m is a solution of Δ m x G m (x, y) = δ(x − y), x,y ∈ Ω, subject to zero Dirichlet conditions for x ∈ ∂Ω. Here δ is the Dirac function.
where c(m, n) does not depend on Ω.

Boundedness of Solutions to the Polyharmonic Equation in Arbitrary Domains
Let Ω be an arbitrary bounded domain in R n and suppose n > 2m + 2 if m > 2 and n ≥ 8 if m = 2.

Problem 33. Prove or disprove that the variational solution of the equation
This is true for the dimensions listed at the end of the previous subsection (see [67]) and fails for the operator [78].

Polyharmonic Capacitary Potentials
Let m = 1, 2, . . .. The m-harmonic capacity of a compact set F in R n is defined by was introduced in [50] (see also [53]) and proved to be useful in the theory of higher order elliptic equations. It is well-known that cap m (F ) = 0 for all compact sets F if 2m ≥ n. Let n > 2m. One of the definitions of the potential-theoretic Riesz capacity of order 2m is as follows: The capacities cap m (F ) and R 2m (F ) are equivalent, that is their ratio is bounded and separated from zero by positive constants depending only on n and m (see [57,62] and [71,Section 13.3]). The minimizer in R 2m is a Riesz potential of a measure whereas that in cap m is a Riesz potential of a distribution. The last minimizer will be denoted by U F . By [67], for the dimensions n = 5, 6, 7 if m = 2 and n = 2m+1, 2m+2 if m > 2, the inequalities 25 Page 20 of 44 hold for all x ∈ R n \F . In general, the constant 2 cannot be replaced by 1.

Problem 34.
Is it possible to extend (5.13) for dimensions n ≥ 8 if m = 2, and n ≥ 2m + 3 for m > 2? Another question: is the constant 2 in (5.13) best possible?

Subadditivity of the Polyharmonic Capacity
Let cap m (F ) and R 2m (F ) be the capacities defined by (5.11) and (5.12). By the classical potential theory, the Riesz capacity R 2m is upper subadditive. However, the following problem is open.

Problem 35.
Prove or disprove that the m-harmonic capacity cap m is upper subadditive.

Normal Derivative of a Harmonic Function at a Boundary Point
Let a δ-neighborhood of the origin O ∈ ∂Ω be given by the inequality x n > ϕ(x ), where ϕ is a Lipschitz function of the variable x ∈ R n−1 , ϕ(0) = 0. We assume that the function Consider also the condition |ξ|<δ ϕ(ξ) d ξ |ξ| n < ∞, (5.15) where the integral is improper.
The following property of harmonic functions appeared in mathematical literature at the end of the XIX century: if a harmonic function takes its minimum value at O, then its inner normal derivative at O is strictly positive.

Problem 36.
Prove that under the a priori requirement (5.14), the condition (5.15) is necessary and sufficient for the validity of the just formulated property of harmonic functions.
Note that (5.15) is much weaker than the Dini condition δ 0 ω(r) r dr < ∞ which previously appeared in the same context (see [2], where other references can be found).
Remark. One possible approach is to use the asymptotic formula for harmonic functions near the Lipschitz boundary established in [30].

Essential Spectrum of a Perturbed Polyharmonic Operator
Consider the selfadjoint operator L in L 2 (R n ) generated by the differential expression where q is a locally integrable nonnegative function in R n .

Problem 37.
Let m > 1 and 2m ≤ n. Is it true that either the essential spectrum of L coincides with the half-axis λ ≥ 0, or the point λ = 0 does not belong to the spectrum of L?

Singularity of Green's Function for Second Order Elliptic Equations with Lower Order Terms It is well-known that Green's function G(x, y) of the uniformly elliptic operator
with measurable bounded coefficients is equivalent to |x − y| 2−n near the point y (see [45,99]).

Generic Degenerating Oblique Derivative Problem
Oblique derivative problem consists in determining solutions of the second order elliptic equation in a domain Ω ⊂ R n subject to the boundary condition where l denotes a field of unit vectors on ∂Ω. The corresponding boundary pseudodifferential operator is elliptic if and only if l is nowhere tangent to ∂Ω.
In the non-elliptic case the oblique derivative problem is called degenerate. Around 1970 Arnold [3] stressed the importance of the so-called generic case of degeneration, where the vector field l is tangent to ∂Ω on a submanifold of codimention 1 and is not transversal to this submanifold (see also his well-known book [4,Section 29]).
In [4, Section 29] Arnold writes: "One of the simplest problems of such a calculation of infinite codimensions corresponding to kernels and cokernels consisting of functions on manifolds of different dimensions is the oblique derivative problem. If we consider this problem on the sphere bounding an n-dimensional ball, a vector field tangent to the n-dimensional ambient space is given. A function harmonic in the ball is to be determined whose derivative in the direction of the field is equal to a given boundary function. We consider, for example, the case n = 3. In this case, a generic field is tangent to the sphere on some smooth curve. There are singular points on this curve where the field is tangent to the curve itself. The structure of the field in the neighborhood of each of these singular points is standard: it can be proved that for any n of a generic field in the neighborhood of every point of the boundary, the field is given, in an appropriate coordinate system, by a formula of the form The oblique derivative problem apparently has to be stated according to the following scheme. The manifold of tangency of the field with the boundary, the manifold of the tangency of the field with the tangency manifolds, etc. divide the boundary into parts of various dimensions. On some of these parts of the boundary, conditions have to be given: on some other parts, the boundary function itself has to satisfy certain conditions for the existence of the classical solution of the problem." In 1972 [59] I published a result related to the generic degeneration, still the only known one. I assumed that there are smooth manifolds Γ 0 ⊃ Γ 1 . . . ⊃ Γ s of dimensions n − 2, n − 3, . . . , n − 2 − s such that l is tangent to Γ j exactly at the points of Γ j+1 , whereas l is nowhere tangent to Γ s . A local model of this situation is given by the following: Some function spaces of the right-hand side and solutions ensuring the unique solvability of the problem are found in [59]. The success was achieved by a choice of weight functions in the derivation of a priori estimates for the solution. Additionally, I proved that the inverse operator of the problem is always compact on L p (Γ), 1 < p ≤ ∞. It turned out that the tangent manifolds of codimension greater than one do not influence the correct statement of the problem, contrary to Arnold's expectations.
However, the following problem is still unsolved.

Problem 39.
Describe the asymptotic behaviour of solutions near the points of tangency of the field l. In particular, it seems challenging to find asymptotics near the point of Γ 1 whose neighbourhood contains both points of entrance and exit of the field l with respect to Γ. More generally, it would be interesting to obtain regularity results for strong solutions of the generic degenerating problem.

Matrix Generalization of the Oblique Derivative Problem
Let Ω be a domain in R n . Consider the Laplace equation Δ u = 0 in Ω, where u is a k-dimensional vector field, and add the condition where {A i } 1≤i≤n is a collection of k × k matrix function.

Problem 40.
Develop a theory of solvability of the boundary value problem above in the case when the corresponding pseudodifferential operator on ∂Ω is not elliptic. An interesting particular case is the boundary condition where a and b are real vector fields and u is a complex-valued scalar function.

Coercive Weighted Estimates for Elliptic Equations in Domains with
Inner Cusps Consider the Dirichlet problem for an arbitrary elliptic equation of order 2m in a domain Ω ⊂ R n with inner cusp. Two-sided estimates for derivatives of order 2m of solutions were obtained in [32,Chapter 10] with n > 2m + 1.

Problem 41.
Obtain estimates of the same nature for n ≤ 2m + 1.

L p -Dissipativity of the Lamé Operator
Consider the Dirichlet problem with zero boundary condition for the Lamé operator L defined in the previous section.
In [9, Chapter 3], we prove that for n = 2 the operator L is L p -dissipative if and only if For n = 3 this inequality is necessary for the L p -dissipativity of L and there are some sufficient conditions. However, the following problem is unsolved.

Problem 43.
Prove or disprove that (5.18) is sufficient for the L p -dissipativity of L in the three-dimensional case.

Generalized Maximum Principle for the Lamé System in a Domain with Non-Lipschitz Graph Conical Vertex
Let Ω be a domain with compact closure and whose boundary ∂Ω is smooth outside one point. We assume that near this point ∂Ω coincides with a cone, not necessarily a Lipschitz graph. We consider the Dirichlet problem where ν < 1. The case ν = 1/2 corresponds to the Stokes system, when the condition div u = 0 becomes explicit. By [81] for n = 3 (see also [33,Sections 3.6 and 5.5]) and [13] for n ≥ 3, the generalized maximum principle u C(Ω) ≤ C g C(∂Ω) (5.19) holds provided ν ≤ 1/2. The proof fails for ν > 1/2, which gives rise to the following Problem 44. Prove (5.19) under the assumption 1/2 < ν < 1.

Boundedness of the Displacement Vector in the Traction Problem in Domains with Non-Lipschitz Graph Conical Vertices
Let Ω be a domain in R 3 with ∂Ω smooth outside one point, the vertex of a non-Lipschitz graph cone. Consider the Neumann problem for the linear elasticity system where f ∈ C ∞ 0 (Ω), u is the displacement vector,

Problem 45. Prove or disprove that the displacement vector is uniformly bounded.
In case of Lipschitz graph domains, the positive answer was given in [12,29], and [11].

Comparison of Martin's and Euclidian Topologies
In [65] a description of the Martin boundary and of the minimal positive harmonic functions were given without proofs for a class of domains in R n . The description just mentioned is as follows.
Let Ω be a bounded domain in R n with Euclidean boundary ∂Ω, O ∈ ∂Ω, and let ∂Ω\{O} be a C 2 manifold. Take spherical coordinates (ρ, τ, α) with origin O (ρ ≥ 0, |τ | ≤ π/2, α ∈ S n−2 ) and consider the case when Ω is obtained by rotating about the axis |τ | = π/2 a domain ω contained in the half-plane {(ρ, τ ) : ρ > 0, |τ | < π/2}, which is given near ρ = 0 by an inequality of the form |τ − ψ(ρ)| < Θ(ρ), so that the boundary of Ω near O is made up of two tangential components under certain conditions on ψ and Θ. The necessary and sufficient condition for O to give rise to exactly one Martin boundary point is the divergence of the integral In other words, (5.21) is equivalent to the existence of the limit where G is Green's function and x, y, z are points in Ω. For me, the period of time when I worked on the topic, was difficult and I was not able to prepare a detailed exposition. Even later I did not return to the area so that [65] became a collection of unproved lemmas. Since it is impossible for me to do this job in the future, I dare to propose it as an unsolved Problem 46. Justify the criterion (5.21) and prove assertions on the Martin boundary formulated in [65].

Nonlinear Singularities at the Vertex of a Cone
Let Δ p be the p-Laplace operator u → div(|∇ u| p−2 grad u).

Problem 47.
No analogue of (6.1) is known for higher order nonlinear equations where a α are positive homogeneous vector-valued functions. The same applies elliptic systems of the second order.

Nonlinear Boundary Value Problem with Infinite Dirichlet Data
Let Ω be a bounded smooth domain in R n and let Q denote a function on Ω × R n , positive homogeneous of degree p ∈ (1, ∞), Q(x, ξ) > 0 for x ∈ Ω, ξ = 0. By Δ p we denote the p-Laplace operator.

Problem 48.
Prove that there exists a unique positive λ such that the Dirichlet problem is uniquely solvable up to a constant term.
Remark. The pair (u, λ) should solve the variational problem It is obvious that for p = 2 and Q(ξ) = |ξ| 2 , (6.2) is the classical variational principle for the first eigenvalue of the Laplacian [96].

Singularities of Solutions to the Neumann Problem for a Semilinear Equation
Let the planar domain Ω have an angle with vertex O. The Neumann problem is considered in [28], where an asymptotic formula for an unbounded solution is given under the assumption of the positivity of the quadratic form Roughly speaking, the solution has a log log r −1 singularity. This asymptotic behaviour fails without positivity of the quadratic form. Indeed, the function u(x) = log r −1 + cos θ satisfies Δ u + cos θ 1 + sin 2 θ |∇ u| 2 = 0 on R 2 + as well as zero Neumann condition. This example suggests the following problem.

Problem 49.
Describe the asymptotic behaviour of solutions to (6.3) without assumption of positivity of the quadratic form (6.4).

Positive Solutions of a Non-Linear Dirichlet Problem with Zero Boundary Data
Let Ω be a bounded planar domain and let ∂Ω be smooth except for an angular point O.

Positive p-Harmonic Functions in a Cone
Let u denote a positive locally bounded solution of the equation Δ p u = 0 in a cone vanishing at the boundary.

Problem 51. Prove that
where λ > 0 and C is an arbitrary constant.
The existence of solutions appearing in the right-hand side of (6.5) was studied in [37] and [103]. If the cone is the half-space {x : x n > 0}, formula (6.5) becomes u(x) = Cx n .

Phragmen-Lindelöf Principle for the p-Laplace equation
By [58], bounded solutions u(x) of the p-Laplace equation, 1 < p < n, with zero Dirichlet data in a δ-neighborhood of a point O ∈ ∂Ω, admit the pointwise majorant This assertion was established for the linear uniformly elliptic equation with measurable bounded coefficients in [55].

Isolated Singularity of Solution to a Nonlinear Elliptic System
In [16], J. Frehse noticed that the elliptic system with smooth nonlinearity has the weak discontinuous solution u 1 = cos log r, u 2 = sin log r, Problem 53. Is the right-hand side of (6.8) the only possible isolated singularity of solutions to system (6.7) at the origin?

Poisson Type Solutions of Riccati's Equation with Dirichlet Data
Let Ω be a smooth, bounded planar domain and let the origin O be a point of ∂Ω. Consider the equation Δ u + α u 2 x + β u 2 y = 0 in Ω, where α and β are smooth positive functions given on Ω. We assume that the solution u satisfies u = 0 on ∂Ω\{O}.

Problem 54.
Prove that either u = 0 on ∂Ω, or u has a logarithmic singularity at O and depends upon an arbitrary constant.
A formal asymptotic expansion of u near O can be found in [92], also for more general equations. in the half-cylinder {(x, t) : x ∈ Ω, t > 0}, where A is a positive-definite matrix and Ω is a bounded domain in R n . Assume that

Consider the equation
If A is the unit matrix, the replacement of u by log v with v > 0 shows that at infinity either u vanishes exponentially, or u behaves as t at any positive distance from the boundary.

Problem 55. Show that the same alternative holds if A is not the unit matrix. Describe an asymptotic behaviour of unbounded solutions. What happens if
A is not positive-definite?

Two Inequalities Related to the Heat Equation
Let Ω be an open subset of R n . By [49] and [71, Section 2.5.2], the inequality where μ is a non-negative measure in Ω, holds for all u ∈ C ∞ 0 (Ω) if and only if Here cap(F, Ω) is the relative harmonic capacity of a compact subset F of Ω with respect to Ω Inequality (7.1) is important in particular in the theory of the Schrödinger operator Δ + μ, see [49] and [71, Section 2.5].
The following inequality finds applications in the theory of the Cauchy-Dirichlet problem for the operator in the cylinder Ω × (0, T ), where μ is a measure defined on this cylinder: where L −1,2 (Ω) is the dual space of L 1,2 0 (Ω). A necessary condition of type (7.2) obviously holds with the parabolic capacity studied in [97]. Most probably, it is also sufficient.

Trace Space for the Maximal Operator Generated by the Wave Operator
Let Ω be a domain in R 1 × R n and let x,t = ∂ 2 /∂t 2 − Δ x . By denote the maximal operator generated by x,t , i.e. the closure of x,t defined on C ∞ (Ω) in the norm Problem 57. Describe the space of boundary traces for functions in the domain of .

Characteristic Problem for Nonlinear Hyperbolic Operators
In [106], the authors study the initial value problem for a hyperbolic linear equation of order 2m, when the initial surface has characteristic points on some compact set Γ. It is proved that the problem is well posed if the set Γ is free from the last initial condition D 2m−1 t u = f . In the case m = 1 a similar result was later obtained in [22].

Problem 58. Obtain a similar result without linearity assumption for the hyperbolic operator.
The only result in this direction known to me is in [6]. It concerns the global characteristic Cauchy problem (Goursat problem) for the nonlinear wave equation. The boundary data in [6] are prescribed on the light cone with two singularities representing both past and future.

Boundary Traces of Functions in the Domain of a Maximal Differential Operator with t-Dependent Coefficients
Let P (D t , D x ) be an arbitrary differential operator acting on functions defined on the half-space R n x ∈ R n , t > 0} and let P max be the corresponding maximal operator. The domain of P max is the completion of the space of functions smooth on R n + and vanishing at infinity in the norm In [18,Chapter 2], an algebraic characterization of boundary traces of functions in the domain of P max was found.
Problem 59. Describe traces on the hyperplane t = 0 of functions in the domain of P max assuming that P is an arbitrary differential operator of the first order in t with coefficients depending on t. This is only an example which points to the possible development of the theory in [18] for differential operators with coefficients depending only on t.

Differential Operators Acting in Pairs of Sobolev Spaces
Obviously, a differential operator of order h with variable coefficients maps the Sobolev space L l,p (R n ) into L l−h,p (R n ) if the coefficients are multipliers in the proper pairs of Sobolev spaces. This statement can be inverted for some classes of differential operators as shown in [88,Section 10.1.1]. However, the following counterexample proposed in [88,Section 10.1.2] shows that this is not a general property of differential operators. The coefficient a of the operator where W l,2 = L l,2 ∩ L 2 , need not be a multiplier from W 1,2 (R n ) into L 2 (R n ). This gives rise to the following Problem 60. Find a condition on the function a, necessary and sufficient for the validity of the inequality

Existence of a Well-Posed Boundary Value Problem for for General
Differential Operators in L p (Ω) Hörmander (1955) showed the existence of a well-posed boundary value problem for every differential operator with constant coefficients in an arbitrary bounded domain. Equivalently, he proved the estimate where u is an arbitrary function in C ∞ 0 (Ω). Problem 61. Does (7.6) hold with L p (Ω) instead of L 2 (Ω)?

Existence of a Well-Posed Boundary Value Problem for Unbounded Do-
mains Estimate (7.6) fails for some unbounded domains which gives rise to Problem 62. Let L(D) be a differential operator in R n with constant coefficients. Characterise the domains satisfying (7.6).  One can even ask the same question for special operators. For instance, it seems interesting to characterize unbounded domains for which the inequalities Here and ∂ xn −Δ x are the wave and heat operators.

Hölder Regularity of Solutions to the n-Dimensional Dirichlet-Stokes Problem
Consider the Dirichlet problem for the Stokes system where Ω is a polyhedron in R n , n ≥ 3, and v ∈ L 1,2 (Ω).
Remark. The starting point could be the following property obtained in [13]: for any solution of the form |x| λ ψ(x/|x|) with Reλ > 1−n/2 one has Reλ > 0.

Differentiability of Solutions to the Dirichlet Problem for the Equations of Hydrodynamics in Convex Domains
Consider the Dirichlet problem for the Stokes system −ν Δv + ∇ p = f in Ω, div v = 0 in Ω, v = g on ∂Ω. (8.1) The following result is obtained in [30, Section 6.3].
Let Ω be a bounded convex two-dimensional domain and let f ∈ L q (Ω), for some q > 2. Then v ∈ C 0,1 (Ω), i.e. v is Lipschitz and where C depends only on Ω.
The following similar result for the Navier-Stokes system was also established in [30,Section 6.3].
Problem 64. Show that the above two facts hold with q > 2 replaced by q > 3 for the three-dimensional case.

Boundedness of Solutions to the Dirichlet Problem for the Stokes System in Arbitrary 3D Domains
Let Ω be an arbitrary bounded three-dimensional domain. Consider the Dirichlet problem (8.1).

Problem 65.
Prove or disprove that the variational solutions are uniformly bounded in Ω.

Resolvent L p -Estimates for the Dirichlet-Stokes Operator
Let R λ be the resolvent operator of the Dirichlet problem where λ ∈ R 1 and Ω is a bounded domain in R n .
where c is a constant which does not depend on λ.

Non-Uniqueness for the Stationary Navier-Stokes System with Dirichlet Data
Let Ω be a simply connected bounded domain in R 3 . It is well known [40] that the Dirichlet problem for the stationary Navier-Stokes system has at most one variational solution provided the Reynolds number is sufficiently small. It seems probable that the uniqueness fails for large Reynolds numbers.  This problem is classical but seems half-forgotten to me. Therefore, I include it in the present list.

Well-Posed Neumann-Kelvin Problem for a Surface-Piercing Body
When solving the seakeeping problem with forward speed which is called Neumann-Kelvin problem, one looks for a velocity potential u(x, y, z) defined on a lower half-space outside the wetted part S of a body moving in the xdirection. The normal derivative of u is prescribed on S and the condition should be satisfied on the flat surface F outside the body (see [38,Chapter 8]). This classical boundary value problem is in an unsatisfactory state both from theoretical and numerical points of view.

Problem 68.
Verify the following conjecture. The Neumann-Kelvin problem becomes Fredholm if the above formulation is completed by prescribing the elevation of the free surface on a part of the water line L, to be more precise, by adding the values of ∂u/∂x at those points of L where the angle between the exterior normal to L and the x-axis is not greater than π/2.
Note that the asymptotics of u near the curve L is not known. However, the existence of ∂u/∂x is suggested by the asymptotic formula (8.8) in [38] obtained for the two-dimensional case.

Solvability of the Two-Dimensional Kelvin-Neumann Problem for a Submerged Cylinder
In [46], the two-dimensional problem on the steady flow of infinite depth about a submerged cylinder is considered and the existence of the unique solution of any velocity v of the undisturbed flow upstream in the case of an arbitrary circular cylinder is proved.
This boundary value problem is stated as follows. One looks for the velocity potential u(x, y) of the steady motion of a heavy ideal incompressible fluid induced by the cylinder with the cross-section Ω which moves uniformly in the x-direction. The function u satisfies where R 2 − = {(x, y) : y < 0} and ν = gv −2 with g being the acceleration due to gravity. In addition, 3) ∂u ∂n = v cos(N, x) on∂Ω, (9.4) where N is the unit normal to ∂Ω directed into Ω. The second relation in (9.3) is equivalent to the absence of waves far upstream.
The result obtained in [46], which is of course rather special, seems, however, to be the only known uniqueness theorem for the boundary value problem just stated not relying on any restrictions on ν.
Hence we naturally arrive at the following Problem 69. Show the unique solvability of (9.1)-(9.4) for a fairly arbitrary domain and for all values of ν.
Note that the proof in [46] is based on the theorem in [105] ensuring the uniqueness of a solution with finite Dirichlet integral under the assumption The proof of uniqueness in [105] concerns the n-dimensional case, n ≥ 2.
For a modern state of the art presentation of the linear theory of water waves see [38].

Counterexample in the Water-Wave Problem for a Submerged Body
Let us use the geometrical assumptions made in Section 48. The boundary value problem of harmonic oscillations of the fluid induced by a submerged body is stated as follows. The function u satisfies (9.1), (9.3), while (9.2) is replaced by ∂ u ∂y − ν u = 0 for y = 0 and a radiation condition at infinity is required (see [38]).
In [64] a class of domains is found for which the problem in question is uniquely solvable for all ν > 0, see also [38,Chapter 1].

Problem 70.
Construct an example of a connected set Ω showing that some restrictions on Ω ensuring solvability are necessary.

Sharp Hardy-Leray Inequality for Divergence-Free Fields
Let u denote a vector field in R n with components in C ∞ 0 (R n ). The following n-dimensional generalization of the one-dimensional Hardy inequality [21] R n |u| 2 |x| 2 dx ≤ 4 (n − 2) 2 R n |∇u| 2 dx (9.5) appears for n = 3 in Leray's pioneering paper on the Navier-Stokes equations [42]. The constant factor on the right-hand side is sharp. Since one frequently deals with divergence-free fields in hydrodynamics, it is natural to ask whether this restriction can improve the constant in (9.5).
It is shown in [10] that this is the case indeed if n > 2 and the vector field u is axisymmetric by proving that the aforementioned constant can be replaced by the (smaller) optimal value 4 (n − 2) 2 1 − 8 (n + 2) 2 (9.6) which, in particular, becomes 68/25 in three dimensions. However, the following problem remains unsolved. Prove or disprove that the constant 4(n − 2) −2 is optimal if u is an arbitrary divergence-free vector field with components in C ∞ 0 (R n ).

A Modified Riemann's Zeta Function Appearing in a Non-Local Parabolic Cauchy Problem
Let C be a unit circle and let u denote a function defined on C. Consider the integral operator A: The spectrum of A is described in [ admits a meromorphic extension to the complex z = t + i τ plane. Study properties of this extension.

Uniqueness Criterion for Analytic Functions with Finite Dirichlet Integral
The following question was raised by Carleson [8]. Suppose that f is analytic in the unit disc U and U |f | p dA < ∞ for some p > 1. Let E be a subset of (−π, π) and suppose that for θ ∈ E, f re iθ → 0 as r → 1.
Of what size must E be to force the conclusion that f is identically zero?
In [8], sufficient conditions for a set of zero length which ensure the uniqueness in the case p = 2 are given. One of the conditions is the positivity of the Riesz potential theoretic capacity R α of a certain positive order α.
Maz'ya and Havin [75] described a class of uniqueness sets which is not included into Carleson's. To state our theorem, denote by E a Borel subset of ∂U and let Δ be a set of non-overlapping open arcs δ ⊂ ∂U . Let |δ| be the length of δ and put E δ := E ∩ δ. Suppose p ∈ (1, 2) and δ∈Δ |δ| log |δ| c(δ) = −∞, (10.1) where c(δ) := cap p (E δ ) is the capacity of E δ defined by (6.6). Then E is a uniqueness set for A 1,p , the class of all functions f analytic in U with f ∈ L p (U ). This theorem is also valid for p = 2 with a slightly different meaning of c(δ) due to the peculiarity of dimension two: it is now the cap 2 -capacity of E δ with respect to the disc 2 d δ , where d δ is a disc contained in E δ . This capacity can be expressed by the logarithmic capacity of E δ or by its transfinite diameter. The case (p, ∞) can also be included, but it is not interesting. It is not hard to construct a set E satisfying (10.1) whose Riesz capacities R α (E) of any order α > 0 vanish. Thus our theorem enlarges the class of uniqueness sets for A 1,2 found in [8].
However, the following problem is still open.
Problem 73. Give a complete characterization of the uniqueness sets for analytic functions in the class A 1,p (U ).

Hybrid Iterative Methods for Solving Boundary Value Problems
In [34], a mathematical justification of certain new iterative schemes used in solving the Dirichlet and Neumann problems for the Laplace equation is given. These schemes are based on a combination of Green's formula and some numerical methods, FEM, for example, which is applied to some auxiliary mixed boundary value problem on a subset of the original domain. The proofs in [34] rely upon geometrical requirements on the boundary of the domain of strong convexity type which do not seem natural.

Problem 74.
Extend the class of domains with preservation of convergence of the iterative procedures proposed in [34].

Asymptotic Optimization of Multi-Structures
The present section concerns boundary value problems for multi-structures, i.e. domains dependent on small parameters in such a way that the limit region, as parameters tend to zero, consists of subsets of different space dimensions. Asymptotic analysis of physical fields in multi-structures is developed in [31,77] and elsewhere. Direct methods of variational calculus are often ineffective for solving problems of optimal control of multi-structures because of their complicated geometry. However, [77,Remark 4.4] suggests the following promising Problem 75. Apply algebraic optimization methods to asymptotic approximations of fields in 1D-3D multi-structures.
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