On the Existence of L2 Boundary Values of Solutions to an Elliptic Equation

Gushchin, A. K.

doi:10.1134/s0081543819050067

Proc. Steklov Inst. Math.

2019

DOI: 10.1134/s0081543819050067

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On the Existence of L2 Boundary Values of Solutions to an Elliptic Equation

A. K. Gushchin

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Cited by 2 publications

References 37 publications

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The boundary values of solutions of an elliptic equation

Gushchin¹

2019

Sb. Math.

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The paper is devoted to the study of the boundary behaviour of solutions of a second-order elliptic equation. Criteria are established for the existence of a boundary value of a solution of the homogeneous equation under the same conditions on the coefficients of the equation as were used to establish that the Dirichlet problem with a boundary function in , , has a unique solution. In particular, an analogue of Riesz’s well-known theorem (on the boundary values of an analytic function) is proved: if a family of norms in the space of the traces of a solution on surfaces ‘parallel’ to the boundary is bounded, then this family of traces converges in . This means that the solution of the equation under consideration is a solution of the Dirichlet problem with a certain boundary value in . Estimates of the nontangential maximal function and of an analogue of the Luzin area integral hold for such a solution, which make it possible to claim that the boundary value is taken in a substantially stronger sense. Bibliography: 57 titles.

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The boundary values of solutions of an elliptic equation

Gushchin¹

2019

Sb. Math.

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On some properties of elliptic partial differential equation solutions

Gushchin

2022

Int. J. Mod. Phys. A

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This paper is devoted to the study of properties of second-order elliptic equation solutions. The main content of the paper coincides with the report made by the author at the international conference dedicated to the 75th anniversary of I. V. Volovich. The solution behavior near the boundary and the Dirichlet problem formulation, which is closely related to this issue, are studied. At the end of the paper, we will briefly discuss the results obtained in elegant and extremely important works by E. De Giorgi and J. Nash regarding Hölder continuity of the equation solutions within the considered domain. We present results that combine and complement the belonging of the solution to the Hölder and Sobolev spaces. Note that all the concepts and statements under consideration are united by a common approach and are formulated in close terms.

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On the Existence of L2 Boundary Values of Solutions to an Elliptic Equation

Cited by 2 publications

References 37 publications

The boundary values of solutions of an elliptic equation

The boundary values of solutions of an elliptic equation

On some properties of elliptic partial differential equation solutions

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