The Trace Theorem for Anisotropic Sobolev — Slobodetskii Spaces with Applications to Nonhomogeneous Elliptic BVPs
Abstract
In this paper, anisotropic Sobolev — Slobodetskii spaces in poly-cylindrical domains of any dimension N are considered. In the first part of the paper we revisit the well-known Lions — Magenes Trace Theorem (1961) and, naturally, extend regularity results for the trace and lift operators onto the anisotropic case. As a byproduct, we build a generalization of the Kruzhkov — Korolev Trace Theorem for the first-order Sobolev Spaces (1985). In the second part of the paper we observe the nonhomogeneous Dirichlet, Neumann, and Robin problems for p-elliptic equations. The well-posedness theory for these problems can be successfully constructed using isotropic theory, and the corresponding results are outlined in the paper. Clearly, in such a unilateral approach, the anisotropic features are ignored and the results are far beyond the critical regularity. In the paper, the refinement of the trace theorem is done by the constructed extension.
DOI 10.14258/izvasu(2018)4-19
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Antontsev S.N., Diaz J.I., and Shmarev S. Energy Methods for Free Boundary Problems. Applications to Nonlinear PDEs and Fluid Mechanics. - Boston, 2002.
DiBenedetto E. Degenerate Parabolic Equations. - New York, 1993.
V´azquez J.L. Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type. - New York, 2006.
Antontsev S.N. and Kuznetsov I.V. Singular perturbations of forward-backward p-parabolic equations. // JEPE. - 2016. - Vol. 2.
Antontsev S.N. and Kuznetsov I.V. Existence of entropy measure-valued solutions for forwardbackward p-parabolic equations. // Sib. Elect. Math.
Rep. - 2017. - Vol. 14.
Kuznetsov I.V. and Sazhenkov S.A. Anisotropic vanishing diffusion method applied to genuinely nonlinear forward-backward ultraparabolic equations. // Sib. Elect. Math. Rep. -2018. — Vol. 15.
Lions J.-L. and Magenes E. Problemi ai limiti non omogenei (III) // Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e serie. — 1961. — V. 15, № 1-2. (In Italian.)
Lions J.-L. and Magenes E. Problemes aux limites non homogenes et applications, vol. 1 et 2. — Paris, 1968. (In French.)
Slobodetskii L.N. Generalized S.L. Sobolev spaces and their application to boundary value problems for partial differential equations. // Scientific Notes of Leningrad State Ped. Inst. — 1958. — Issue 197. (In Russian.)
Steinbach O. Numerical Approximation Methods for Elliptic Boundary Value Problems, Finite and Boundary Elements. — New York, 2003.
Lions J.-L. Quelques Methodes de Resolution des Problemes aux Limites Non Lineaire. — Paris, 1969. (In French.)
Kruzhkov S. and Korolev A. Towards a theory of embedding of anisotropic functional spaces. // Dokl. Acad. Nauk. SSSR. — 1985. — V. 285. (In Russian.)
Ohno M., Shizuta Y., and Yanagisawa T. The trace theorem on anisotropic Sobolev spaces // Tohoku Math. J. — 1994. — V. 46.
Meyries M. and Schnaubelt R. Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights. // Journal of Functional Analysis. — 2012. — V. 262.
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