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Elliptic boundary-value problems in some distribution spaces of generalized smoothness

Anna Anop, Aleksandr Murach

TL;DR

The paper extends solvability theory for elliptic boundary-value problems to Sobolev spaces of generalized smoothness $H^{s,\varphi}_{p}$, refined by a slowly varying function $\varphi\in\Upsilon$ and constructed through a blend of complex interpolation with a function parameter and quadratic interpolation between Hilbert spaces. It establishes localization and trace results to Besov spaces $B^{s-1/p,\varphi}_{p}$, proves a Fredholm framework with stable kernel/index, and provides exact sufficiency (and necessity) conditions for prescribed generalized or classical smoothness, including local regularity. It further develops parameter-elliptic theory, proving isomorphisms and uniform a priori estimates in parameter-dependent norms, with implications for parabolic problems and spectral analysis. Overall, the work broadens elliptic theory to refined scales, enabling precise smoothness control and robust analysis across parameter regimes.

Abstract

We build a solvability theory of elliptic boundary-value problems in normed Sobolev spaces of generalized smoothness for any integrability exponent $p>1$. The smoothness is given by a number parameter and a supplementary function parameter that varies slowly at infinity. These spaces are obtained by a combination of the methods of the complex interpolation with number parameter between Banach spaces and the quadratic interpolation with function parameter between Hilbert spaces applied to classical Sobolev spaces. We show that the spaces under study admit localization near a smooth boundary and describe their trace spaces in terms of Besov spaces with the same supplementary function parameter. We prove that a general differential elliptic problem induces Fredholm bounded operators on appropriate pairs of the spaces under study. We also find exact sufficient conditions for solutions of the problem to have a prescribed generalized or classical smoothness on a given set and establish corresponding a priori estimates of the solution. These results are specified for parameter-elliptic problems.

Elliptic boundary-value problems in some distribution spaces of generalized smoothness

TL;DR

The paper extends solvability theory for elliptic boundary-value problems to Sobolev spaces of generalized smoothness , refined by a slowly varying function and constructed through a blend of complex interpolation with a function parameter and quadratic interpolation between Hilbert spaces. It establishes localization and trace results to Besov spaces , proves a Fredholm framework with stable kernel/index, and provides exact sufficiency (and necessity) conditions for prescribed generalized or classical smoothness, including local regularity. It further develops parameter-elliptic theory, proving isomorphisms and uniform a priori estimates in parameter-dependent norms, with implications for parabolic problems and spectral analysis. Overall, the work broadens elliptic theory to refined scales, enabling precise smoothness control and robust analysis across parameter regimes.

Abstract

We build a solvability theory of elliptic boundary-value problems in normed Sobolev spaces of generalized smoothness for any integrability exponent . The smoothness is given by a number parameter and a supplementary function parameter that varies slowly at infinity. These spaces are obtained by a combination of the methods of the complex interpolation with number parameter between Banach spaces and the quadratic interpolation with function parameter between Hilbert spaces applied to classical Sobolev spaces. We show that the spaces under study admit localization near a smooth boundary and describe their trace spaces in terms of Besov spaces with the same supplementary function parameter. We prove that a general differential elliptic problem induces Fredholm bounded operators on appropriate pairs of the spaces under study. We also find exact sufficient conditions for solutions of the problem to have a prescribed generalized or classical smoothness on a given set and establish corresponding a priori estimates of the solution. These results are specified for parameter-elliptic problems.

Paper Structure

This paper contains 4 sections, 16 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Spaces of generalized smoothness
  3. Applications to elliptic problems
  4. Applications to parameter-elliptic boundary-value problems

Key Result

Proposition 2.1

Let $s,\varepsilon,\delta\in\mathbb{R}$, $\varepsilon,\delta>0$, and $\varphi\in\Upsilon$. Define the interpolation function parameter as follows: Suppose that $\Omega$ satisfies Omega-assumption. Then up to equivalence of norms.

Theorems & Definitions (34)

  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof : Proof of Theorems $\ref{['th-C-interp-Sobolev']}$ and $\ref{['th-C-interp-Sobolev-got']}$.
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • Theorem 2.6
  • Proposition 2.8
  • proof : Proof of Theorems $\ref{['th-Besov-properties']}$ and $\ref{['th-C-interp-Besov']}$.
  • ...and 24 more