Local boundedness, maximum principles, and continuity of solutions to infinitely degenerate elliptic equations with rough coefficients / Lyudmila Korobenko, Cristian Rios, Eric Sawyer, Ruipeng Shen.

Format:

Book

Author/Creator:

Korobenko, Lyudmila, 1985- author.

Rios, Cristian, 1969- author.

Sawyer, E. T. (Eric T.), 1951- author.

Shen, Ruipeng, author.

Series:

Memoirs of the American Mathematical Society ; no. 1311.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1311

Language:

English

Subjects (All):

Differential equations, Elliptic--Numerical solutions.

Differential equations, Elliptic.

Function spaces.

Physical Description:

vii, 130 pages : illustrations ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [2021]

Summary:

"We obtain local boundedness and maximum principles for weak subsolutions to certain infinitely degenerate elliptic divergence form inhomogeneous equations, and also continuity of weak solutions to homogeneous equations. For example, we consider the family {f[sigma]}[sigma]>0 with f[sigma] (x) = e -( 1 [pipe]x[pipe] ) [sigma] , -[infinity] <x< [infinity], of infinitely degenerate functions at the origin, and show that all weak solutions to the associated infinitely degenerate quasilinear equations of the form divA (x, u) gradu = [phi] (x), A (x, z) ̃̃ In-1 0 0 f (x1) 2 , with rough data A and [phi], are locally bounded for admissible [phi] provided 0 <[sigma]< 1. We also show that these conditions are necessary for local boundedness in dimension n [greater than or equal to] 3, thus paralleling the known theory for the smooth Kusuoka-Strook operators [partial derivative]2 [partial derivative]x2 1 + [partial derivative]2 [partial derivative]x2 2 +f[sigma] (x) 2 [partial derivative]2 [partial derivative]x2 3 . We also show that subsolutions satisfy a maximum principle under the same restriction on the degeneracy. Finally, continuity of solutions is derived in the homogeneous case [phi] [triple bar] 0 under a more stringent assumption on the degeneracy, namely that f [greater than or equal to] f3,[sigma] for 0 <[sigma]< 1 where f3,[sigma] (x) = [pipe]x[pipe] (ln ln ln 1 [pipe]x[pipe] ) [sigma] , -[infinity] <x< [infinity]. As an application we obtain weak hypoellipticity (i.e. smoothness of all weak solutions) of certain infinitely degenerate quasilinear equations in the plane [partial derivative]u [partial derivative]x2 + f (x, u (x, y))2 [partial derivative]u [partial derivative]y2 = 0, with smooth data f (x, z) f3,[sigma] (x) where f (x, z) has a sufficiently mild nonlinearity and degeneracy. In order to prove these theorems, we first establish abstract results in which certain Poincarè and Orlicz Sobolev inequalities are assumed to hold. We then develop subrepresentation inequalities for control geometries in order to obtain the needed Poincarè and Orlicz Sobolev inequalities"- Provided by publisher.

Contents:

DeGiorgi iteration, local boundedness, maximum principle and continuity

Organization of the proofs

Local boundedness

Maximum principle

Continuity

Infinitely degenerate geometries

Orlicz norm Sobolev inequalities

Geometric theorems.

Notes:

"January 2021, volume 269, number 1311 (second of 7 numbers)."

Includes bibliographical references.

ISBN:

9781470444013

1470444011

OCLC:

1228926894