Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains

This is a survey of results mostly relating elliptic equations and systems of arbitrary even order with rough coefficients in Lipschitz graph domains. Asymptotic properties of solutions at a point of a Lipschitz boundary are also discussed.


Introduction
The fundamental role in the theory of linear elliptic equations and systems is played by results on regularity of solutions up to the boundary of a domain. The following classical example serves as an illustration.
Consider the Dirichlet problem ∆u = 0 in Ω, where Ω is a bounded domain with smooth boundary and Tr u stands for the boundary value (trace) of u. Let u ∈ L 1 p (Ω), 1 < p < ∞, that is Moreover, the harmonic extension u of f ∈ B 1−1/p p (∂Ω) belongs to L 1 p (Ω) and the norm of the gradient ∇u in L p (Ω) is equivalent to the above seminorm in B 1−1/p p (∂Ω).
This fact highlights the following topics of interest: • Replace the domain with smooth boundary by a domain in more general class and study the effect of irregularities of ∂Ω.
• Replace ∆ by a more general elliptic operator with variable coefficients, and study the impact of low regularity assumptions on the coefficients.
• Understand the correlation between the smoothness of data and the smoothness of solutions.
Another theme, somehow related to these, is: • Describe the local behaviour of solutions near a boundary or interior point of the domain Ω.
In the present paper we survey results in the directions just mentioned. Most of them were obtained during the last decade and concern elliptic equations and systems of arbitrary order. Special attention is paid to the Stokes system. The selection of topics is partly influenced by our involvement in their study. Here is the plan of the article.
Section 1 is dedicated to weak solutions with Besov boundary data, with coefficients of differential operators and the unit normal to the boundary in classes close to BM O. Results of a similar nature for the Stokes system are discussed in Section 2. In Section 3, strong solutions in Sobolev spaces are considered. Here sharp additional conditions on the Lipschitz boundary are reviewed. In particular, in Subsection 3.2 we speak about strong solvability of the Stokes system. In Section 4, asymptotic formulas for solutions of the Dirichlet problem near an isolated point of the Lipschitz boundary and at a point in the domain are dealt with.
1 Weak solvability of higher order elliptic systems with coefficients close to BMO in Lipschitz domains

Background
The present section is mostly based on results of the paper by V. Maz'ya, M. Mitrea and T. Shaposhnikova [MMS1]. We start with mentioning earlier works.
The basic case of the Laplacian in arbitrary Lipschitz graph domains in R n , is treated in the work of B. Dahlberg and C. Kenig [DK], in the case of L p -data, and D. Jerison and C. Kenig [JK], in the case of data in the Besov space B s p with 0 < s < 1. The local regularity in the Sobolev class W 2 p of solutions to second order equations with coefficients in V M O ∩ L ∞ was established by F. Chiarenza, M. Frasca, P. Longo [CFL].
In spite of substantial progress in recent years, there remain many basic open questions for higher order equations, even in the case of constant coefficient operators in Lipschitz domains. One significant problem is to determine the sharp range of p's for which the Dirichlet problem for strongly elliptic systems with L p -boundary data is well-posed. In [PV], J. Pipher and G. Verchota have developed a L p -theory for real, constant coefficient, higher order systems when p is near 2, i.e., 2 − ε < p < 2 + ε with ε > 0 depending on the Lipschitz character of Ω. On p. 2 of [PV] the authors ask whether the L p -Dirichlet problem for these operators is solvable in a given Lipschitz domain for p ∈ (2 − ε, 2(n−1) n−3 + ε), and a positive answer has been given by Z. Shen in [Sh]. Let us also mention the work [AP] of V. Adolfsson and J. Pipher who have dealt with the Dirichlet problem for the biharmonic operator in arbitrary Lipschitz domains and with data in Besov spaces, as well as [Ve] where G. Verchota formulates and solves a Neumann-type problem for the bi-Laplacian in Lipschitz domains and with boundary data in L 2 . In [MMT] D. Mitrea, M. Mitrea, and M. Taylor, treat the Dirichlet problem for strongly elliptic systems of second order in an arbitrary Lipschitz subdomain Ω of a (smooth) Riemannian manifold and with boundary data in B s p (∂Ω), when 2 − ε < p < 2 + ε and 0 < s < 1. We mention some recent results on the second order elliptic equations and systems with coefficients in V M O due to G. Di Fazio [DF], L. Caffarelli and I. Peral [CP], B. Stroffolini [St], D. Guidetti [Gu], for higher order equations see also D. Palagachev and L. Softova [PS].

Domains and function spaces
Let us turn to the article [MMS1]. We make no notational distinction between spaces of scalar-valued functions and their natural counterparts for vector-valued functions.
Recall that a domain Ω is called Lipschitz graph if its boundary can be locally described by means of (appropriately rotated and translated) graphs of real-valued Lipschitz functions.
It is shown by S. Hofmann, M. Mitrea, and M. Taylor in [HMT] that Ω is a Lipschitz graph domain if and only if it has finite perimeter in the sense of De Giorgi (see [DG1], [Fe], [BM]) and (i) there are continuous (or, equivalently, smooth) vector fields that are transversal to the boundary and (ii) the necessary condition ∂Ω = ∂Ω is fulfilled.
Everywhere in this section we assume that Ω is a Lipschitz graph domain in R n , with compact closure Ω and with outward unit normal ν = (ν 1 , ..., ν n ). Let m be an integer. Consider the operator The coefficients A αβ are square matrices with measurable, complex-valued entries, for and such that the coercivity condition holds for all complex vector-valued functions V ∈ C ∞ 0 (Ω). Let U belong to the usual Sobolev space W m p (Ω). It is natural to take (Ω) is defined as the space of vector-valued functions for which (Ω), the traces of D α U, 0 ≤ |α| ≤ m − 1, exist in B s p (∂Ω), where s := 1 − a − 1/p, 0 < s < 1 (see E. Gagliardo [Gag] for a = 0 and S. Uspenskiȋ [Usp]). Recall that f ∈ L p (∂Ω) belongs to B s p (∂Ω) if and only if The above definition takes advantage of the Lipschitz manifold structure of ∂Ω which allows one to define smoothness spaces of index less than 1. The approach is no longer effective when the order of smoothness exceeds 1.
Let us illustrate the necessity of working with boundary data different from those in spaces of traces of usual Sobolev spaces by considering Dirichlet problem for the biharmonic operator U ∈ W 2 2 (Ω), ∆ 2 U = 0 in Ω, Tr U = g 0 on ∂Ω, ν, Tr [∇U] = g 1 on ∂Ω.
Since this function does not belong to W 2 2 (S), there is no solution in this space. A new point of view has been introduced by H. Whitney in [Wh] who considered higher order Lipschitz spaces on arbitrary closed sets. An extension of this circle of ideas pertaining to the full scale of Besov and Sobolev spaces on irregular subsets of R n can be found in the book [JW] by A. Jonsson and H. Wallin. The authors of [MMS1] further refined this theory in the context of Lipschitz domains. The description of higher order Besov spaces on the boundary of a Lipschitz domain Ω ⊂ R n in [MMS1] runs as follows.
In order to formulate the trace and extension theorem for the spacesḂ m−1+s p (∂Ω), we first give its analogue for lower smoothness spaces which is essentially due to S. Uspenskiȋ [Usp].
For higher smoothness see the following assertion which is Proposition 7.3 in [MMS1].
Proposition 1 For 1 < p < ∞, −1/p < a < 1 − 1/p, s := 1 − a − 1/p ∈ (0, 1) and m ∈ N, define the higher order trace operator where the traces in the right-hand side are taken in the sense of Lemma 1. Then (1.12) is a well-defined, linear, bounded operator, which is onto and has V m,a p (Ω) as its null-space. Moreover, it has a bounded, linear right-inverse, i.e., there exists a linear, continuous operator (1.14) such thatḟ Now, a necessary condition for the boundary data {g k } 0≤k≤m−1 in the Dirichlet problem (1.2) is that This family {g k } is organized as a Banach space, and is denoted byẆ m−1+s p (∂Ω).
The space takes a particularly simple form when m = 2. To describe it we need the notation for the tangential derivative ∂/∂τ jk given by 1.17) and the tangential gradient on the surface ∂Ω Then, for each Lipschitz graph domain Ω ⊂ R n and each 1 < p < ∞, s ∈ (0, 1), This has been conjectured to hold (when s = 1 − 1/p) by A. Buffa and G. Geymonat on p. 703 of [BG].

Formulation of the Dirichlet problem
Broadly speaking, there are two types of questions pertaining to the well-posedness of the Dirichlet problem in a Lipschitz domain Ω for a divergence form strongly elliptic system of order 2m with boundary data inẆ m−1+s p (∂Ω).
Question I. Granted that the coefficients of L exhibit a certain amount of smoothness, identify the indices p, s for which this boundary value problem is well-posed.
Question II. Alternatively, having fixed the indices s and p, characterize the smoothness of ∂Ω and of the coefficients of L for which the aforementioned problem is wellposed.
Both questions are discussed in [MMS1].
B. Dahlberg (see [Dah1] has constructed a domain Ω ⊂ R n , bounded with C 1boundary, and a function f ∈ C ∞ (Ω) such that, for each p ∈ (1, ∞), where ∂ j = ∂/∂x j . In other words, Dahlberg's domain Ω has the property that for each p ∈ (1, ∞), the Poisson problem for the Laplacian with homogeneous Dirichlet boundary conditions fails to be well posed, in the sense that it is no longer reasonable to expect solutions with two derivatives in L p (Ω). A fundamental issue raised by this counterexample: Identify those Sobolev-Besov spaces within which the natural correlation between the smoothness of the data and that of the solutions of the Dirichlet problem is preserved when the domain in question is allowed to have a minimally smooth boundary.
Consider the Dirichlet problem for the operator (1.1) (Ω) then, necessarily, and, moreover, For the results in the converse direction, the main hypothesis in [MMS1] requires that, at small scales, the so called local mean oscillations of the unit normal to ∂Ω and of the coefficients of the differential operator L(X, D X ) are not too large, relative to the Lipschitz constant of the domain Ω, the ellipticity constant of L(X, D X ), and the indices of the corresponding Besov space.
By the local mean oscillation of F ∈ L 1 (Ω) we understand where {B ε } Ω stands for the family of balls of radius ε centered at points of Ω and the barred integrals denote the mean values. Similarly, the local mean oscillation of where {B ε } ∂Ω is the collection of n-dimensional balls of radius ε with centers on ∂Ω.
Note that smallness of the local mean oscillation {ν} Osc(∂Ω) does not imply smallness of the Lipschitz constant. Indeed, let 1.4 Solvability of the Dirichlet problem in W m,a p The main result in [MMS1] runs as follows.
Let Ω ⊂ R n be a bounded Lipschitz domain whose Lipschitz constant is does not exceed M , and assume that the operator L(X, D X ) of order 2m is strongly elliptic, and has bounded, measurable complex coefficients.
Theorem 1 There exists a positive constant C, depending only on M and the ellipticity constant of L, such that: For each p ∈ (1, ∞), s ∈ (0, 1) and a := 1 − s − 1/p, the Dirichlet problem (1.20) For second order operators the factor s(1 − s) in the last inequality can be removed. Furthermore, there exists C = C(∂Ω, A, p, s) > 0 such that (1.21) The next assertion, obtained in [MMS1], is a byproduct of the proof of Theorem 1.
Theorem 2 Let Ω ⊂ R n be a bounded Lipschitz domain whose Lipschitz constant does not exeed M , and assume that the operator L(X, D X ) of order 2m is strongly elliptic, and has bounded, measurable (complex) coefficients. Then there exists ε, depending only on M and the ellipticity constant of L, such that: For each p ∈ (1, ∞), s ∈ (0, 1) and a := 1 − s − 1/p, the Dirichlet problem for L with F ∈ V −m,a p (Ω) and g : Recently M. Agranovich [Ag1] obtained this type of results (|p − 2| is small and 0 < s < 1) for both Dirichlet and Neumann problems for a subclass of strongly elliptic systems with Douglis-Nirenberg structure in bounded Lipschitz domains. The regularity results in [Ag1] concern solutions in spaces of Bessel potentials H σ p and Besov spaces B σ p with coefficients of differential operators satisfying the uniform Lipschitz condition. The approach is based on regularity methods due to Savaré as well as on author's developement of interpolation theory of spaces H σ p and B σ p with σ of arbitrary sign (see [Ag2] for more details), where essential role is played by an extension operator from Ω to R n introduced by V. Rychkov [Ry]. To be specific, Agranovich considered a Douglis-Nirenberg system with the principal part L 0 whose entries are given by The coefficients a j,k α,β (x) are complex-valued and the formal self-adjointness of the operatot L j,k is not assumed. The principal symbol of the system, i. e. the matrix L 0 (x, ξ) with entries L j,k (x, ξ) is subordinate to the condition of strong ellipticity: where Λ(ξ) is the diagonal matrix with entries |ξ| 2mj on the main diagonal, j = 1, . . . , l, and C is a positive constant. The use the Savaré method requires the additional condition for any numbers ζ k α at all points x ∈ Ω. Let BMO and VMO stand, respectively, for the space of functions of bounded mean oscillations and the space of functions of vanishing mean oscillations (considered either on Ω, or on ∂Ω). It can be proved that where the distance is taken in BMO. Thus the small oscillation condition introduced in Theorem 1 holds if This is the case if, e.g., [ν] BMO + [A αβ ] BMO is sufficiently small hence, trivially, if ν ∈ VMO(∂Ω) and A αβ belong to VMO(Ω), irrespective of p, s, L and Ω.
Other examples of domains satisfying the hypotheses of Theorem 1 are: Lipschitz domains with a sufficiently small Lipschitz constant, relatively to the exponents p and s. In particular, Lipschitz polyhedral domains with dihedral angles sufficiently close, depending on p and s, to π.
The innovation in [MMS2] that allows to consider boundary data in higher-order Besov spaces, is the systematic use of weighted Sobolev spaces. In relation to the standard Besov scale, we would like to point out that Of course, W m,a p (Ω) is just the classical Sobolev space W m p (Ω) when a = 0. In the context of Theorem 1, for a Lipschitz ∂Ω, This generalizes the trace and extension result by S. Uspenskiȋ [Usp] according to which if 1 < p < ∞ and 0 < s < 1, uniformly for u harmonic in the upper-half space.
Of course, condition (1.3) ensures that the left-hand side of (1.20) is always finite but it is its actual size which determines whether for a given pair of indices s, p, the problem (1.2), (1.4), (1.16) is well-posed. Note that the maximum value that the right-hand side of (1.20) takes for 0 < s < 1 and 1 < p < ∞ occurs precisely when p = 2 and a := 1−s−1/p = 0. As (1.20) shows, the set of pairs (s, 1/p) ∈ (0, 1)×(0, 1) for which (1.2) is well-posed in the context of Theorem 1 exhausts the entire square (0, 1) × (0, 1) as the distance from ν and the A αβ 's to VMO tends to zero (while the Lipschitz constant of Ω and the ellipticity constant of L stay bounded). That the geometry of the Lipschitz domain Ω intervenes in this process through a condition such as (1.20) confirms a conjecture made by P. Auscher and M. Qafsaoui in [AQ].
Theorem 2 can be viewed as a far reaching extension of a well-known theorem of N. Meyers, who has treated the case m = 1, l = 1 in [Mey]. The example given in Section 5 of [Mey] shows that the inclusion of p into a small neighborhood of 2 is a necessary condition, even when ∂Ω is smooth, if the coefficients A αβ are merely bounded. For higher order operators we make use of an example due to V. Maz'ya [Maz1] (cf. also the contemporary article by E. De Giorgi [DG2]). Specifically, when m ∈ N is even, consider the divergence-form equation where L 4 is the fourth order operator Obviously, the coefficients of L 4 (X, D X ) are bounded, and if the parameters a, b, c ∈ R, a > 0, are chosen such that b 2 < ac then L along with ∆ 1 2 m−1 L 4 ∆ 1 2 m−1 are strongly elliptic. Now, if W s p denotes the usual L p -based Sobolev space of order s, it has been observed in [Maz1] that the function U(X) := |X| θ+m−2 ∈ W m 2 (Ω) has Tr U ∈ C ∞ (∂Ω) and is a weak solution of (1.22) for the choice the strong ellipticity condition is satisfied and θ = θ(ε) becomes 2 − n/2 + n ε 1/2 /2 4(n − 1) 2 + ε 1/2 .
However, U ∈ W m p (Ω) if and only if p < n/(2 − θ(ε)), and the bound n/(2 − θ(ε)) approaches 2 when ε → 0. An analogous example can be produced when m > 1 is odd, starting with a sixth order operator L 6 (X, D X ) from [Maz1]. In the above context, given that W 1 n (Ω) ֒→ VMO(Ω), it is significant to point out that both for the example in [Mey], when n = 2, and for (1.22) when n ≥ 3, the coefficients have their gradients in weak-L n , yet they fail to belong to W 1 n (Ω). We conclude this subsection with a remark pertaining to the presence of lower order terms. More specifically, granted Theorem 1, a standard perturbation argument (cf., e.g., [H]) proves the following. Assume that where the principal part of A(X, D X ) satisfies the hypotheses made in Theorem 1 and the lower order terms are bounded. Then, assuming that (1.20) holds, the Dirichlet problem (1.19) is Fredholm with index zero, in the sense that the operator is so. Furthermore, the estimate holds for any solution U ∈ W m,a p (Ω) of (1.19).

Comments on the proof of Theorem 1
One difficulty linked with the case m > 1 arises from the way the norm in ( 1.27) behaves under a change of variables destined to flatten the Lipschitz surface ∂Ω. When m = 1, a simple bi-Lipschitz changes of variables such as Ω ∋ (X ′ , X n ) → (X ′ , X n − ϕ(X ′ )) ∈ R n + will do, but matters are considerable more subtle in the case m > 1. The extension operator used in [MMS1] was introduced by J. Nečas (in a different context; cf. p. 188 in [Nec]) and then rediscovered by V. Maz'ya and T. Shaposhnikova in [MS1] (see also [MS2], and later by B. Dahlberg, C. Kenig J. Pipher, E. Stein and G. Verchota (cf. [Dah2] and the discussion in [DKPV]), and S. Hofmann and J. Lewis in [HL].
The extension operator in question is introduced in the following way. Fix a smooth, radial, decreasing, even, non-negative function ζ in R n−1 such that ζ(t) = 0 for |t| ≥ 1 and acting on functions ϕ from L 1,loc (R n−1 ).
The two estimates below provide useful properties of the operator T .

Another ingredient in the proof of Theorem 1 is deriving estimates for
i.e. a unique solution of the boundary-value problem where y ∈ R n + is regarded as a parameter. The methods employed in earlier works are based on explicit representation formulas for G(x, y) and cannot be adapted to the case of non-symmetric, complex coefficient, higher order systems. The approach in [MMS1] consists of proving that the residual part R(x, y) with respect to ∂R n + .

Stokes system
By B s p,q (R n ) we denote the space of functions in R n having the finite norm The Besov scale B s p,q (Ω) is defined by restricting the (tempered) distributions from the corresponding spaces in R n to the open set Ω. Also, B s p,q (∂Ω) stands for the Besov class on the Lipschitz manifold ∂Ω, obtained by transporting (via a partition of unity and pull-back) the standard scale B s p,q (R n−1 ).

Weak solvability in Besov and Triebel-Lizorkin spaces
Consider the Stokes system in an arbitrary bounded Lipschitz domain Ω ⊂ R n , n ≥ 2, When ∂Ω is sufficiently smooth (at least of the class C 2 ), the problem (2.2) was studied in many papers, first in Sobolev spaces with an integer amount of smoothness by V.A. Solonnikov [Sol], L. Cattabriga [Cat], R. Temam [Tem], Y. Giga [Gig], R. Dautray and J.-L. Lions [DL], among others. This has been subsequently extended by C. Amrouche and V. Girault [AG] to the case when ∂Ω ∈ C 1,1 and, further, by G.P. Galdi, C.G. Simader and H. Sohr [GSS] when ∂Ω is Lipschitz, with a small Lipschitz constant. The case when Ω is a polygonal domain in R 2 , or a polyhedral domain in R 3 also has a reach history. An extended account of this field of research can be found in V. Kozlov, V. Maz'ya and J. Rossmann's monograph [KMR], which also contains references to earlier work. Among recent publications we mention the paper by V. Maz'ya and J. Rossmann [MR1] as well as their book [MR2]. Lipschitz and C 1 subdomains of Riemannian manifolds were treated in the paper [DMi] by M. Dindoš and M. Mitrea, and the paper [MT] by M. Mitrea and M. Taylor.
The principal result in [MMS2] on solutions of the problem (2.2) holds under the mild condition on the normal ν to the boundary of the Lipschitz graph domain and runs as follows.
Theorem 3 Assume that Then there exists δ > 0 which depends only on the Lipschitz character of Ω and the exponent p, with the property that if {ν} Osc(∂Ω) < δ, then the problem (2.2) is well-posed (with uniqueness modulo locally constant functions in Ω for the pressure).
There exists a finite, positive constant C = C(Ω, p, q, s, n) such that where the infimum is taken over all locally constant functions c in Ω.
Moreover, analogous well-posedness result holds on the Triebel-Lizorkin scale (for its definition see, for example, the book by T. Runst and W. Sickel [RS]), i.e. for the problem This time, in addition to the previous conditions imposed on the indices p, q, it is also assumed that p, q < ∞.
It should be noted that conditions (2.4) describe the largest range of indices p, q, s for which the Besov spaces B s p,q (∂Ω) can be meaningfully defined on the Lipschitz manifold ∂Ω.
Regarding the nature of the main result, note that no topological restrictions are imposed on the Lipschitz domain Ω (in particular, the boundary can be disconnected). This is significant since the approach is via boundary layer potentials, whose invertibility properties are directly affected by topological nature of the domain. That difficulty is overcome by using certain mapping properties of the hydrostatic layer potentials: single layer potential S and double layer potential D λ for the velocity and single layer potential Q and double layer potential P λ for the presure. We list their properties in the following theorem.

Dirichlet data in L p (Ω)
We say a few words about the Dirichlet problem for the multi-dimensional Stokes system with L p boundary data which is not touched upon in [MMS1]. Following the paper by M. Dindoš and V. Maz'ya [DMa], we shall speak about both the Lamé system (with the Poisson ratio α < 1/2) and the Stokes system (with α = 1/2).
Let us consider a bounded domain Ω in R n , n ≥ 3, and the system ∆u − ∇π = 0, div u + (1 − 2α)π = 0 in Ω (2.6) complemented by the condition and the class of solutions is described by the inclusion u * ∈ L p (∂Ω). (2.8) The boundary values Tr u are understood in nontangential sense, that is in the sense of the limit Tr u(x) = lim y→x,y∈Γ(x) u(y), over a collection of interior nontangential cones Γ(x) of some aperture and height and vertex at x ∈ ∂ω, and u * is the classical nontangential maximal function defined as One says that problem (2.6), (2.7) is L p solvable if for all vector fields h ∈ L p (∂Ω) there is a pair (u, π) satisfying (2.6) -(2.8) and moreover, for some C > 0 independent of h, the estimate u * Lp(∂Ω) ≤ C h Lp(∂Ω) holds. Furthermore, the problem (2.6), (2.7) is said to be solvable for continuous data if, for all h ∈ C(∂Ω) the vector field u belongs to C(Ω) and the estimate u * C(Ω) ≤ C h C(∂Ω) holds.
The L p solvability of the Dirichlet problem for the Stokes system is established by Z. Shen [Sh] for all p ∈ (2−ε(Ω), ∞] provided Ω is a three-dimensional Lipschitz graph domain. In [DMa], the problem (2.6), (2.7) is considered on domains in R n , n ≥ 3, with isolated conical singularity (not necessary a Lipschitz graph) and the authors prove its solvability for all p ∈ (2 − ε(Ω), ∞] as well as its solvability in C(Ω) for the data in C(∂Ω). This seems to be a strong indication that the range p ∈ (2 − ε(Ω), ∞] should hold also for Lipschitz graph domains. However, there is no such a result for n > 3.

Lipschitz continuous solutions
We cite a regularity result for solutions of the Dirichlet problem for the Stokes system in a plane convex domain obtained by V. Kozlov and V. Maz'ya in [KM3]: where f ∈ W −1 2 (Ω) and (u, π) ∈W 1 2 (Ω) × L 2 (Ω).

Theorem 5
Let Ω be a bounded convex two-dimensional domain and let f ∈ L q (Ω) for some q > 2. Then the velocity vector u ∈W 1 2 (Ω) admits the estimate where C depends only on Ω.
A direct consequence of this result for the nonlinear Navier-Stokes system is as follows.
A result of the same nature was obtained by V. Maz'ya in [Maz3] for solutions of the Neumann problem for the Poisson equation in arbitrary convex n-dimensional domain. Let Ω be a bounded convex domain in R n and let W l p (Ω) stand for the Sobolev space of functions in L p (Ω) with distributional derivatives of order l in W l p (Ω). By L p,⊥ (Ω) and W l p,⊥ (Ω) one means the subpaces of functions in L p (Ω) and W l p (Ω) subject to Ω v dx = 0.
Theorem 6 Let f ∈ L q,⊥ (Ω) with a certain q > n and let u be the unique function in W 1 2 (Ω), also orthogonal to 1 in L 2 (Ω) and satisfying the Neumann problem −∆u = f in Ω, where ν is the unit outward normal to ∂Ω and the problem (2.11) is understood in the variational sense. Then the solution u ∈ W l p,⊥ (Ω) of the problem (2.11) satisfies the estimate ∇u L∞(Ω) ≤ C f Lq(Ω) . (2.12) As a particular case of a result obtained by A. Cianchi and V. Maz'ya in [CM] for a class of nonlinear equations, one can replace the L q -norm on the right-hand side by the Lorentz norm f L n,1 (Ω) which is the best possible majorant among those formulated in terms of Lorentz spaces. Similar facts for Lamé and Stokes systems with boundary conditions different from those of Dirichlet are unknown.

Preliminaries
In this section we give applications of Sobolev multipliers to the question of higher regularity in fractional Sobolev spaces of solutions to boundary value problems for higher order elliptic equations in a Lipschitz domain. Since the sole Lipschitz graph property of Ω does not guarantee higher regularity of solutions, we are forced to select an appropriate subclass of Lipschitz domains whose description involves a space of multipliers. For domains of this subclass we develop a solvability and regularity theory analogous to the classical one for smooth domains. We also show that the chosen subclass of Lipschitz domains proves to be best possible in a certain sense.
Let Ω be a domain in R n with compact closure. Throughout this section we assume that for any point of the boundary ∂Ω there exists a neighbourhood U and a Lipschitz function ϕ such that (3.1) Let B l−1/p p (R n−1 ), l = 1, 2, . . . , denote the completion of the space C ∞ 0 (R n−1 ) in the norm Replacing R n−1 by ∂Ω one arrives at the definition of the space B l−1/p p (∂Ω).

By the space of Sobolev multipliers
(3. 2) The notation M W l p (Ω) is used for h = l. Properties and applications of Sobolev multipliers are studied in [MS2].
We introduce the essential norm of a function γ ∈ M (W h p (Ω) → W l p (Ω)): where {T } is the set of all compact operators: W h p (Ω) → W l p (Ω). Analytic two-sided and one-sided estimates for the norm (3.2) and the essential norm (3.3) can be found in [MS2].

Subclasses of Lipschitz graph domains
Let Ω be a bounded Lipschitz graph domain. We introduce the class M l−1/p p (l = 2, 3, . . . ) of boundaries ∂Ω, satisfying the following condition. For every point of ∂Ω there exists an n-dimensional neighborhood in which ∂Ω is specified (in a certain According to Corollary 4.3.8 in [MS2], for p(l − 1) > n we have where B r (x) is a ball in R n−1 with radius r and center x. Therefore, the classes M l−1/p p and B l−1/p p coincide for p(l − 1) > n.
Here B ε is the ball with centre at O and radius ε, c is a constant which depends only on l, p, n, and Proposition 3 and properties of the capacity lead to the following sufficient condition formulated in terms of the (n − 1)-dimensional Lebesgue measure m n−1 .
Corollary 2 (i) If n > p(l − 1) and Corollary 3 Let p(l − 1) ≤ n and let Ω be a bounded Lipschitz graph domain with Further, let ∂Ω be locally defined in Cartesian coordinates by y = ϕ(x), where ϕ is a function with a Lipschitz constant less than c δ. Then ∂Ω ∈ M l−1/p p (δ).
Setting q = ∞ in Corollary 3, one obtains the simple sufficient condition for the inclusion into M l−1/p p (δ) formulated in terms of the modulus of continuity ω l−1 (t) of ∇ l−1 ϕ:

General elliptic boundary value problem
Consider either scalar or square matrix differential operators . We assume that the operators P , tr P 1 , . . . , tr P m form an elliptic boundary value problem in every point O ∈ ∂Ω with respect to the hyperplane y = 0 and that P is an elliptic operator in Ω.
The next result is proved essentially in the same manner as Theorem 14.3.1 in [MS2].
Theorem 7 (i) Suppose that for any neighbourhood U ⊂ R n there exist operators with constant coefficients such that {P U ; Tr P U j } is the operator of an elliptic boundary value problem in the half-space {x ∈ R n−1 , y ≥ 0}. ( where l is integer, l ≥ 2m, 1 < p < ∞. The constant δ here and elsewhere is supposed to be small in comparison with the norms of the inverse operators {P U ; Tr P U j } −1 for all U . Further, let the coefficients a αj satisfy similar conditions with 2m replaced by k j . is Fredholm. In particular, for all u ∈ W l p (Ω) the a priori estimate holds; the last norm in the right-hand side can be omitted in the case of a unique solution.
Note that even the roughest sufficient condition (3.7) on the domain for (iii) in Theorem 7 to hold is sharp. Let ω be an increasing function in C[0, 1] such that ω(0) = 0, It was shown in Section 4.4.3 of [MS2] that one can construct a function ϕ on R n−1 such that (i) the continuity modulus of ∇ l−1 ϕ does not exceed c ω with c = const; Given ϕ, one can construct a bounded domain Ω in R n such that the Neumann problem ∆v − v = g in Ω, ∂v/∂ν = h on ∂Ω (3.12) with g ∈ W l−2 p (Ω) and h ∈ B l−1−1/p p (∂Ω) may fail to be solvable in W l p (Ω). Next we describe conditions on the coefficients of (3.8) which are equivalent to those formulated in Theorem 7 and follow from the results in [MS2], Ch. 7. In its formulation, we use the notion of capacity of a compact set in R n defined similarly to (3.5).
Corollary 4 Conditions (3.9) in Theorem 7 can be stated as follows: (i) The coefficients a α with |α| = 2m are in the class W l−2m p (Ω) if p(l − 2m) > n and satisfy the inequality Various other sufficient conditions for (3.9) follow from the results in [MS2].

Dirichlet problem in terms of traces
Let us first consider the Dirichlet problem as a particular case of the general boundary value problem dealt with in Subsection 3.3. We write the scalar or square matrix elliptic operator P in the form (3.13) Suppose that the coefficients A αβ are in C l−m (Ω), l ≥ m, and that the Gårding inequality Re Ω |α|=|β|=m (3.14) holds for u ∈ C ∞ 0 (Ω). Assume that Ω is a Lipschitz graph domain. We introduce a sufficiently small finite open covering {U } ofΩ and a corresponding partition of unity {ζ U }. Let The Dirichlet boundary conditions will be prescribed by the operators We give a formulation of the Dirichlet problem. Let us look for a function u ∈ W l p (Ω) such that P u = f in Ω, Tr P j u = f j on ∂Ω, j = 1, . . . , m , where f and f j are functions in W l−2m p (Ω) and B l+1−j−1/p p (∂Ω) respectively. Here is the principal result relating the problem (3.15) borrowed from Subsection 14.5.4 in [MS2].
An example in Subsection 14.6.1 of [MS2] shows that for p(l − 1) ≤ n and for m = 1 the condition ∂Ω ∈ M l−1/p p (δ) in part (α) of Theorem 8 cannot be replaced by the assumption that ∂Ω belongs to the class M l−1/p p ∩ C l−1 . To be precise, a domain Ω is constructed with ∂Ω ∈ M 3/2 2 ∩ C 1 for which the problem is solvable in W 2 2 (Ω) not for all f ∈ L 2 (Ω). This means that the smallness of the seminorm ∇ϕ MB 1/2 2 (R n−1 ) in the definition of M 3/2 2 (δ) is essential for the solvability of problem (3.16) in W 2 2 (Ω). The next assertion, which follows directly from the Implicit Function Theorem 9.5.2 in [MS2], shows that the condition ∂Ω ∈ B l−1/p p with p(l − 1) > n is necessary for the solvability of problem (3.15) in W l p (Ω) for the operator P of higher than second order.
Theorem 10 Let ∂Ω be in the class C l−2,1 and let l be integer, l ≥ 2m, p(l − 1) ≤ n, 1 < p < ∞, and m > 1. If there exists a solution u ∈ W l p (Ω) of problem (3.17), then ∂Ω ∈ B l−1/p p . A similar result for the second order operator P in Subsection 14.6.2 of [MS2] runs as follows.
Theorem 11 Let l be integer, l ≥ 2, 1 < p < ∞, m = 1, and P 1 ≤ 0. Let Ω be a domain with ∂Ω ∈ C 1 and let the normal to ∂Ω satisfy the Dini condition. If, for a nonpositive function f ∈ C ∞ 0 (Ω), there exists a solution u ∈ W l p (Ω) of the problem P u = f in Ω, Tr u = 0 , (3.18) Note that the convergence requirement (3.7), the heaviest assumption on Ω made in Corollary 3, is in a sense a sharp condition for solvability of the Dirichlet problems (3.17) and (3.18) in W l p (Ω). The corresponding domain is constructed with the help of the same function ω as in the case of the Neumann problem (3.12) (see Example 15.6.1 in [MS2]).
Example 15.5.2 from [MS2] shows that surfaces in the class M l−1/p p (δ) with p(l − 1) < n may have conic vertices if n > p(l − 1) and s-dimensional edges if s < n − p(l − 1).
Suppose that for any point O ∈ ∂Ω there exists a neighborhood U such that U ∩ Ω is C ∞ -diffeomorphic to the domain i.e. the dimensions of boundary singularities are at most n − 1 − s. Then (3.4) is equivalent to ∇ϕ MB l−1−1/p p (R n−1−s ) ≤ c δ and, in particular, it takes the form Next we state two corollaries of (3.20).
Note that under the conditions of the last theorem, the operator is continuous.
4 Asymptotic formulas for solutions to elliptic equations

Asymptotics of solutions near Lipschitz boundary
Results of a new type were obtained by V. Kozlov and V. Maz'ya in [KM3] for solutions of the Dirichlet problem for higher order elliptic equations in Lipschitz graph domains. We mean an explicit description of the asymptotic behaviour of solutions near a point O of the Lipschitz boundary.
Consider the special Lipschitz graph domain where ϕ(0) = 0 and ϕ has a small Lipschitz constant. The authors of [KM3] study solutions of arbitrary strongly elliptic equation of order 2m with constant complexvalued coefficients with zero Dirichlet data on (B 3 ∩ ∂G)\O, where B ρ = {x : |x| < ρ} and ∂ x = (∂ 1 , . . . , ∂ n ). It is assumed that the operator L has no lower-order terms and the coefficient in front of ∂ 2m n is equal to (−1) m .
One of the results in [KM3] is the existence of a solution U ∈ W m 2 (G) of the homogeneous equation (4.1)) which admits the asymptotic representation Here ε is a poisitive constant, and E is the Poisson solution of the equation L(∂ x )E(x) = 0 in the upper half-space R n + which is positive homogeneous of degree m − n and is subject to the Dirichlet conditions on the hyperplane x n = 0 where δ is the Dirac function.
It is also shown that a multiple of U is the main term in the asymptotic representation of an arbitrary solution u if both u and f are subject to mild growth conditions near O.
Solutions with the weakest possible singularity at O are studied in [KM3] as well. The authors construct a solution U of the homogeneous equation (4.1) which is subject to the asymptotic formula The asymptotic formulas (4.2) and (4.3) can be simplified under additional conditions on κ(ρ). Let, in particular, Then, in the special case of the polyharmonic equation any solution u satisfying |u(x)| = O(|x| m−n+1−ε ) is subject to the following alternatives: either

Asymptotics of solutions to equations with discontinuous coefficients near a smooth boundary
Proofs of the just mentioned results in [KM3] rely upon the papers [KM1], [KM2] on the asymptotic formulas for solutions to the Dirichlet problem for arbitrary even order 2m strongly elliptic equations of divergence form near a point O at the smooth boundary. It is required only that the coefficients of the principal part of the operator have small oscillation near this point, and the coefficients in lower order terms are allowed to have singularities at the boundary.
We say a few words on the proof of asymptotic formulas in [KM1], [KM2]. The elliptic equation is transformed to a first-order evolution system with the matrix whose entries are partial differential operators on the hemisphere with time dependent coefficients. Thus, the question of asymptotics of solutions to the original Dirichlet problem is reduced to the study of the long-time behaviour of solutions of the evolution system just mentioned. The structure of the operator matrix in the system is rather complicated, because it has been obtained from a higher order partial differential equation in the variational form. Moreover, the study of this system is aggreviated by the scantiness of information about the behaviour of the operator matrics at infinity. This difficulty is overcome by a right choice of function spaces, characterizing the solutions and the right-hand side of the evolution system by certain seminorms depending on time. To obtain an asymptotic formula for the solution, the authors apply a particular spectral splitting of the system into one-dimensional and infinite-dimensional parts. A general asymptotic theory of differential equations with operator coefficients in Banach spaces which is the basis of [KM1], [KM2] is developed in [KM].
As an illustration, we describe a corollary of the main result in [KM1] concerning second order equations. Consider the uniformly elliptic equation in Ω (4.4) complemented by the Dirichlet condition where Ω is a domain in R n with smooth boundary. We assume that elements of the n × n-matrix A(x) are measurable and bounded complex-valued functions. One consideres a solution u with a finite Dirichlet integral and suppose, for simplicity, that f = 0 in a certain δ-neighborhood Ω δ = {x ∈ Ω : |x| < δ} of the origin. Further, it is assumed that there exists a constant symmetric matrix A with positive definite real part such that the function is sufficiently small for r < δ.
We introduce the function where |S n−1 | is the Lebesgue measure of the unit sphere in R n , z, ζ = z 1 ζ 1 +. . .+z n ζ n and ν is the interior unit normal at a point O on the boundary of Ω. (For the notation (detA) 1/2 and A −1 x, x n/2 see [H], Sect. 6.2).
The following asymptotic formula for an arbitrary solution of (4.4), (4.5) with finite energy integral is a special case of the main theorem in [KM1]: where C = const and ε is a small positive number.
Using (4.6), it is an easy matter to derive sharp two-sided estimate for the Hölder exponent of u at the origin. Another direct application of (4.6) is the following criterion. Under the condition Re R dx > −∞. (4.8) Needless to say, this new one-sided restriction (4.8) is weaker than the classical Dini condition at the origin. The complementary assumption (4.7) appeared previously in several papers dealing with other problems of the boundary behaviour of solutions to equation (4.4) (see the articles by E. Fabes, D. Jerison, and C. Kenig [FJK], by B.E. Dahlberg [Dah3], by C. Kenig [Ken] et al).
Let v be a solution of the equation (4.4) complemented by the Dirichlet condition v = 0 on ∂Ω\{O}, O ∈ ∂Ω, which has an infinite energy integral and the least possible singularity. We state a particular case of the main theorem in [KM1] which is the following asymptotic representation for v: where C = const and ε is a small positive number.
In general, theorems proved in [KM1] and [KM2] provide asymptotic formulas similar to (4.6) and (4.9) for solutions of the Dirichlet problem for the higher order uniformly elliptic equation with complex-valued coefficients 0≤|α|,|β|≤m The only a priori assumption on the coefficients L αβ is smallness of the function where x ∈ B + δ and L αβ are constants.

Corollaries of the asymptotic formulas in Section 4.1
The last section in [KM3] concerns, in particular, solutions to the Dirichlet problem for elliptic equations of order 2m with constant coefficients in plane domains with a small Lipschitz constant of the boundary as well as arbitrary bounded plane convex domans Ω. Let Ω be a bounded domain in R 2 . Consider a strongly elliptic operator with constant coefficients and denote by w a weak solution to the Dirichlet problem (Ω), this problem is uniquely solvable. let us assume that where ϕ is a Lipschitz function on [−2δ 0 , −2δ 0 ] and ϕ(0) = 0. Note that one does not require the convexity of ϕ.
The next result concerning solutions to problem (4.10), which stems from (4.2), is as follows.
Note that for the operator ∆ m one has ζ ± k = ±i, which implies a = −m/π. The next assertion is a consequence of Theorem 15 when the function ϕ is convex.
Theorem 16 Suppose that the function ϕ describing the domain Ω near the point O is non-negative and convex, and |ϕ ′ (±2δ)| does not exceed a sufficiently small constant l 0 depending on m and the coefficients of L(∂ x ). Furthermore, let f be zero in Ω ∩ B 2δ and let w be a solution of (4.10), which is extended by zero outside Ω. Then (4.13) where δ < δ 0 /8 and A(δ) is given by (4.12).
One of the main results obtained in [KM3] concerns the Green function G L of the Dirichlet problem for the operator L with real coefficients.

Theorem 17
Let Ω be an arbitrary bounded convex domain in R 2 and let the coefficients of L be real. Then, for all x, y in Ω, where C is a positive constant depending on Ω.
The case of complex coefficients is more complicated.
Theorem 18 Let L be an arbitrary strongly elliptic operator with complex coefficients. Suppose that Ω is a bounded convex domain such that the jumps of all angles between the exterior normal vector to ∂Ω and the x-axis be smaller than a constant depending on m and the coefficients of L(∂ x ). Then, for all x, y in Ω, estimate (4.14) holds.
Theorem 17 implies the following regularity result.

Corollary 5
Let Ω be an arbitrary bounded convex domain in R 2 and let the coefficients of L be real. Then the solution w of problem (4.10) with f ∈ W 1−m q (Ω), q > 2, satisfies where the constant C depends on Ω, m, q, and the coefficients of L(∂ x ).
Generally, this assertion does not hold for operators with complex coefficients. More precisely, if there exists an angle vertex on the boundary of a convex domain Ω, one can construct a second order strongly elliptic operator L(∂ x ) with complex coefficients such that the Dirichlet problem (4.10) with f ∈ C(Ω) has a solution with unbounded gradient (see [KMR], Sect. 8.4.3). By Theorem 18, the statement of Corollary 5 for L with complex coefficients holds if the jumps of the normal vector are either absent or small.

Classical asymptotics of solutions near a point of the domain
Now, we present some results borrowed from Sect. 14.9 of the book by V. Kozlov and V. Maz'ya [KM] which are devoted to the asymptotic behaviour of solutions to elliptic equations near an interior point O of the domain. Here a modified Dini-type condition on the coefficients is introduced which guarantees the preservation of the asymptotics of solutions to the main part of the equation with coefficients frozen at O.
and let G denote the Green matrix of this operator, i.e. the solution of the system where I l is the l × l identity matrix and δ is the Dirac function. It is well-known (see F. John's book [Jo]) that G admits the representation G(x) = r 2m−n Q(ω) if 2m ≥ n, n odd, or 2m < n R(x) log r + r 2m−n S(ω) if 2m ≥ n, n even, where Q and S are smooth matrix-functions on the unit sphere in R n and R is a homogeneous polynomial matrix of degree 2m − n. where K r = {x ∈ R n : e −1 r < |x| < r}, and assume that S(r) does not exceed a small positive constant. We shall also use the notation . We formulate three theorems on the asymptotic representation as x → 0 for solutions u ∈ W 2m 2,loc (B r0 \{0}) of Q(x, D x )u = 0 on B r0 \{0} (4.17) satisfying u W 2m 2 (Kr) = O(r k+δ ) (4.18) with some δ > 0 and integer k.
Theorem 19 Let 2m < n and where γ is a positive integer.

22)
where C α = const and G is the Green matrix introduced by (4.16).
The asymptotics (4.23) can be made more precise under the assumption that the operator Q contains no derivatives of order |α| < s, that is The formula (4.23) can be replaced by where v W 2m 2 (Kr) = o(r s | log r| 1−γ ). Theorem 20 Let n be odd, 2m > n and let S be subject to (4.19). Then where c α and C β are constants and v satisfies (4.21). If either k < −1 or k > 2m − n, then the first or the second sum in (4.24) should be omitted.
Theorem 21 Let n be even, 2m > n.

Asymptotics of solutions of the second order equation with square-Dini coefficients
The asymptotic behaviour of solutions near the isolated point O of the domain was recently considered by V. Maz'ya and R. McOwen [MM1], [MM2] for the case of the second order elliptic operator in nondevergence form It is assumed that the coefficients have modulus of continuity ω satisfying the square-Dini condition If the coefficients are real, then, without loss of generality one can put a ij (0) = δ ij .
An important role is played by the function where A z stands for the matrix a ij (z), , is the inner product in R n , and ε is a sufficiently small positive number.
If the coefficients are subject to the usual Dini condition 1 0 ω(t) dt t < ∞, (4.28) then, obviously, there exists a finite limit of I(r) as r → 0, but (4.28) is not necessary for the existence of this limit. In general, under (4.26), I(r) may be unbounded as r → 0, but, clearly, for every λ > 0 there exists C λ such that |I(r)| ≤ λ | log r| + C λ for 0 < r < ε. (4.29) The results in [MM1] and [MM2] are formulated in terms of the L p -means The main theorem in [MM1] can be stated as follows.
Theorem 22 Suppose that where ω satisfies (4.26). For p ∈ (1, ∞) and ε > 0 sufficiently small, there exists a weak solution Z ∈ L p,loc (B ε ) of 1≤i,j≤n ∂ i ∂ j a ij (x) Z = 0 (4.31) satisfying Z(x) = e −I(|x|) (1 + ζ(x)), (4.32) where the remainder term ζ satisfies M p (ζ, r) ≤ c max(ω(r), σ(r)) with σ(r) := Now let us turn to the behaviour of a solution to the homogeneous nondivergence equation with an isolated singularity at x = O. We assume that n > 2. The results obtained in [MM2] imply the existence of the solution Z(x) ∼ |x| 2−n n − 2 e I(|x|) as x → O. (4.36) The behaviour of I(|x|) as x → O not only controls the leading asymptotics of Z(x) but whether we can solve the equation There are three important cases to consider.
In this case Z(x) may be scaled by a constant factor to make it asymptotic to the fundamental solution for the Laplacian. In fact, the distributional equation We see that Z(x) = o(|x| 2−n ) as x → O, and we can solve (4.37) to find C = 0. Thus, in this case we obtain the interesting corollary that Lu = 0 in B ε (4.39) admits a solution u = Z that is quite singular at x = O: |Z(x)| ≥ C λ |x| 2−n+λ for every λ > 0. In particular, local regularity of solutions of the homogeneous equation (4.39) does not hold.