We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain D of Rn for a second order elliptic differential operator A(x, ∂). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ∂) is of Robin type on ∂D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ⊂ ∂D and control the growth of solutions near Y. We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y. Moreover, we prove the completeness of root functions related to L.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price includes VAT (USA)
Tax calculation will be finalised during checkout.
S. Agmon, “On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems”, Comm. Pure Appl.Math. 15, 119 (1962).
M. S. Agranovich, “Elliptic operators on closed manifold,” Current Problems of Mathematics, Fundamental Directions, 63, 5 (VINITI, 1990) [in Russian].
M. S. Agranovich, “On series with respect to root vectors of operators associated with forms having symmetric principal part,” Funct. Anal. Appl. 28 (3), 151 (1994).
M. S. Agranovich, “Non-self-adjoint elliptic problems on non-smooth domains,” Russ. J. Math. Phys. 2 (2), 139 (1994).
M. S. Agranovich, “Spectral problems for second-order strongly elliptic systems in smooth and non-smooth domains,” Russian Math. Surveys 57 (5), 847 (2002).
M. S. Agranovich, “Spectral boundary value problems in Lipschitz domains for strongly elliptic systems in Banach spaces Hs p and Bs p,” Funct. Anal. Appl. 42 (4), 249 (2008).
M. S. Agranovich, “Spectral problems in Lipschitz domains,” Modern Mathematics, Fundamental Trends 39, 11 (2011).
M. S. Agranovich, “Strongly elliptic second order systems with boundary conditions on a non-closed Lipschitz surface,” Funct. Anal. Appl. 45 (1), 1 (2011).
M. S. Agranovich, “Mixed problems in a Lipschitz domain for strongly elliptic second order systems,” Funct. Anal. Appl. 45 (2), 81 (2011).
L. A. Aizenberg and A. M. Kytmanov, “On the possibility of holomorphic continuation to a domain of functions given on a part of its boundary,” Math. Sbornik 182 (4), 490 (1991).
F. E. Browder, “On the eigenfunctions and eigenvalues of the general elliptic differential operator,” Proc. Nat. Acad. Sci. USA 39, 433 (1953).
F. E. Browder, “Estimates and existence theorems for elliptic boundary value problems,” Proc. Nat. Acad. Sci. USA 45, 365 (1959).
F. E. Browder, “On the spectral theory of strongly elliptic differential operators,” Proc. Nat. Acad. Sci. USA 45, 1423 (1959).
V. I. Burenkov, Sobolev Spaces on Domains, Teubner-Texte zurMathematik 137 (B.G. Teubner, Stuttgart, 1998).
N. Dunford and J. T. Schwartz, Linear Operators, Vol. II, Self-Adjoint Operators in Hilbert Space (Intersci. Publ., New York, 1963).
Yu. Egorov, V. Kondratiev, and B. W. Schulze, “Completeness of eigenfunctions of an elliptic operator on a manifold with conical points,” Russ. J. Math. Phys. 8, (3), 267 (2001).
I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Nonself-adjoint Operators in Hilbert Spaces (AMS, Providence, R.I., 1969).
I. Ts. Gokhberg and E. I. Sigal, “An operator generalization of the logarithmic residue theorem and the theorem of Rouché,” Math. Sbornik 13, 603 (1971).
P. Hartman, Ordinary Differential Equation (John Wiley and Sons, New York, 1964).
L. I. Hedberg and T. H. Wolff, “Thin sets in nonlinear potential theory,” Ann. Inst. Fourier (Grenoble) 33 (4), 161 (1983).
L. Hörmander, “Hypoelliptic second order differential equations,” Acta Math. 119, 147 (1967).
M.V. Keldysh, “On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations,” Dokl. AN SSSR 77, 11 (1951) [in Russian].
M. V. Keldysh, “On the completeness of eigenfunctions of some classes of non-self-adjoint linear operators,” Russian Math. Surveys 26 (4), 15 (1971) [in Russian].
J. J. Kohn, “Subellipticity of the ∂-Neumann problem on pseudoconvex domains: sufficient conditions,” Acta Math. 142 (1,2), 79 (1979).
V. A. Kondrat’ev, “Completeness of the systems of root functions of elliptic operators in Banach spaces,” Russ. J. Math. Phys. 6 (10), 194 (1999).
O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Equations of Elliptic Type (Nauka, Moscow, 1973) [in Russian].
J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems und Applications. Vol. 1 (Springer-Verlag, Berlin, 1972).
V. P. Mikhailov, Partial Differential Equations (Nauka, Moscow, 1976) [in Russian].
B.V. Paltsev, “Mixed problemswith non-homogeneous boundary conditions in Lipschitz domains for secondorder elliptic equations with a parameter,” Math. Sbornik, 187 (4), 59 (1996).
B. A. Plamenevskiĭ, Algebras of Pseudodifferential Operators (Nauka, Moscow, 1986) [in Russian].
A. Polkovnikov and A. Shlapunov, “On the spectral properties of a noncoercive mixed problem associated with ∂-operator,” J. Siberian Fed. Uni. 6 (2), 247 (2013) [in Russian].
A. A. Shlapunov, “Spectral decomposition of Green’s integrals and existence of W s,2-solutions of matrix factorizations of the Laplace operator in a ball,” Rend. Sem.Mat. Univ. Padova 96, 237 (1996).
A. A. Shlapunov and N. N. Tarkhanov, “Duality by reproducing kernels,” Int. J. of Math. and Math. Sc. 6, 327 (2003).
A. A. Shlapunov and N. N. Tarkhanov, “On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators,” J. of Differential Equations 255, 3305 (2013).
A. A. Shlapunov and N. N. Tarkhanov, “Sturm-Liouville Problems in Weighted Spaces in Domains with Non-Smooth Edges. II,” Siberian Adv. Math. 26 (1), 30 (2016).
L.N. Slobodetskiĭ, “generalized spaces of S.L. Sobolev and their applications to boundary problems for partial differential equations,” Science Notes of Leningr. Pedag. Institute 197, 54 (1958) [in Russian].
N. Tarkhanov, “On the root functions of general elliptic boundary value problems,” Compl. Anal.Oper. Theory 1, 115 (2006).
A. N. Tikhonov and A. A. Samarskiĭ, Equations of Mathematical Physics (Nauka, Moscow, 1972) [in Russian].
B. L. Van der Waerden, Algebra (Springer-Verlag, Berlin, 1967).
S. Zaremba, “Sur un problème mixte relatif à l’équation de Laplace,” Bull. Acad. Sci. Cracovie, 314 (1910).
Original Russian Text ©A. A. Shlapunov and N. Tarkhanov, 2015, published in Matematicheskie Trudy, 2015, Vol. 18, No. 2, pp. 133–204.
About this article
Cite this article
Shlapunov, A.A., Tarkhanov, N.N. Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II. Sib. Adv. Math. 26, 247–293 (2016). https://doi.org/10.3103/S1055134416040027
- mixed problems
- noncoercive boundary conditions
- elliptic operators
- root functions
- weighted Sobolev spaces