The Generalized Hölder and Morrey-Campanato Dirichlet Problems for Elliptic Systems in the Upper Half-Space

AbstractBy virtue of a weak comparison principle in small domains we prove axial symmetry in convex and symmetric smooth bounded domains as well as radial symmetry in balls for regular solutions of a class of quasi-linear elliptic systems in non-variational form. Moreover, in the two dimensional case, we study the system when set in a half-space.

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Derivative-type ascent formulas for kernels of some half-space Dirichlet problems

The ANZIAM Journal ◽

10.1017/s144618110001186x ◽

2000 ◽

Vol 42 (2) ◽

pp. 185-194

Author(s):

L. R. Bragg

Keyword(s):

Potential Theory ◽

Half Space ◽

Laplace Equation ◽

Bessel Functions ◽

Axially Symmetric ◽

Dirichlet Problems ◽

Symmetric Potential ◽

Derivative Type ◽

Differentiation Formulas

AbstractDerivative-type ascent formulas are deduced for the kernels of certain half-space Dirichlet problems. These have the character of differentiation formulas for the Bessel functions but involve modifying variables after completing the differentiations. The Laplace equation and the equation of generalized axially-symmetric potential theory (GASPT) are considered in these. The methods employed also permit treating abstract versions of Dirichlet problems.

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Another approach of Morrey estimate for linear elliptic equations with partially BMO coefficients in a half space

Filomat ◽

10.2298/fil1804429t ◽

2018 ◽

Vol 32 (4) ◽

pp. 1429-1437

Author(s):

Hong Tian ◽

Shenzhou Zheng

Keyword(s):

Half Space ◽

Elliptic Equations ◽

Elementary Approach ◽

Dirichlet Problems ◽

Spatial Variables ◽

Morrey Estimates ◽

Lp Estimate ◽

Special Weight ◽

Bmo Coefficients ◽

Weak Derivatives

Making use of an elementary approach instead of the weighted Lp estimate with a special weight, we prove global Morrey estimates of the weak derivatives to the Dirichlet problems of linear elliptic equations with small partially BMO coefficients in a half space. Here, the leading coefficients aij(x) are assumed to be merely measurable in one variable, and have small BMO in the remaining spatial variables.

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DIRICHLET PROBLEMS IN A HALF-SPACE OF A HILBERT SPACE

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025702000729 ◽

2002 ◽

Vol 05 (02) ◽

pp. 257-291 ◽

Cited By ~ 2

Author(s):

ENRICO PRIOLA

Keyword(s):

Hilbert Space ◽

Half Space ◽

Regularity Result ◽

Interpolation Theory ◽

Dirichlet Problems ◽

Optimal Regularity ◽

Hölder Continuous ◽

Infinite Dimensional ◽

Order Elliptic Operator ◽

Constant Coefficients

We study a homogeneous infinite dimensional Dirichlet problem in a half-space of a Hilbert space involving a second-order elliptic operator with Hölder continuous coefficients. Thanks to a new explicit formula for the solution in the constant coefficients case, we prove an optimal regularity result of Schauder type. The proof uses nonstandard techniques from semigroups and interpolation theory and involves extensive computations on Gaussian integrals.

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