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Let $Ω \subset ℝ^{n}, n \geq 3$ , and let $p, 1 < p < \infty, p \neq 2$ , be given. In this paper we study the dimension of $p$ -harmonic measures that arise from non-negative solutions to the $p$ -Laplace equation, vanishing on a portion of $\partial Ω$ , in the setting of $δ$ -Reifenberg flat domains. We prove, for $p \geq n$ , that there exists $\tilde{δ} = \tilde{δ} (p, n) > 0$ small such that if $Ω$ is a $δ$ -Reifenberg flat domain with $δ < \tilde{δ}$ , then $p$ -harmonic measure is concentrated on a set of $σ$ -finite $H^{n - 1}$ -measure. We prove, for $p \geq n$ , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$ -harmonic measure...

On the Dirichlet and Neumann problems in multi-dimensional cone

Vladimir Vasilyev (2014)

Mathematica Bohemica

We consider an elliptic pseudodifferential equation in a multi-dimensional cone, and using the wave factorization concept for an elliptic symbol we describe a general solution of such equation in Sobolev-Slobodetskii spaces. This general solution depends on some arbitrary functions, their quantity being determined by an index of the wave factorization. For identifying these arbitrary functions one needs some additional conditions, for example, boundary conditions. Simple boundary value problems,...

On the Dirichlet boundary value problem for nonlinear elliptic partial differential equations in Sobolev power weight spaces

Josef Voldřich (1985)

Časopis pro pěstování matematiky

On the Dirichlet Probelm for Surfaces of Prescribed Mean Curvature.

Mariano Giaquinta (1974)

Manuscripta mathematica

On the Dirichlet problem associated with the Dunkl Laplacian

Mohamed Ben Chrouda (2016)

Annales Polonici Mathematici

This paper deals with the questions of the existence and uniqueness of a solution to the Dirichlet problem associated with the Dunkl Laplacian $Δ_{k}$ as well as the hypoellipticity of $Δ_{k}$ on noninvariant open sets.

On the Dirichlet problem for a degenerate elliptic equation

Jan H. Chabrowski (1987)

Commentationes Mathematicae Universitatis Carolinae

On the Dirichlet problem for a second order elliptic system with degeneration on the entire domain boundary.

Usanetashvili, M. (1999)

Georgian Mathematical Journal

On the Dirichlet problem for linear elliptic equations in plane domains with corners

A. Azzam (1983)

Annales Polonici Mathematici

On the Dirichlet problem for minimal graphs in hyperbolic space.

Fang Hua Lin (1989)

Inventiones mathematicae

On the Dirichlet problem for nonlinear degenerate elliptic equations and applications to optimal control.

Bardi, M., Bottacin, S. (1998)

Rendiconti del Seminario Matematico

On the Dirichlet problem for quasilinear elliptic second order equations with triple degeneracy and singularity in a domain with a boundary conical point.

Borsuk, Michail, Portnyagin, Dmitriy (1999)

Electronic Journal of Differential Equations (EJDE) [electronic only]

On the Dirichlet Problem for Semi-Linear Elliptic Equation with L2-Boundary Data.

J. Chabrowski (1984)

Mathematische Zeitschrift

On the Dirichlet problem for the degenerate real Monge Ampère equation.

Bert G. Wachsmuth (1992)

Mathematische Zeitschrift

On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback

Bao-Zhu Guo, Jun-Min Wang, Cui-Lian Zhou (2008)

ESAIM: Control, Optimisation and Calculus of Variations

We study the dynamic behavior and stability of two connected Rayleigh beams that are subject to, in addition to two sensors and two actuators applied at the joint point, one of the actuators also specially distributed along the beams. We show that with the distributed control employed, there is a set of generalized eigenfunctions of the closed-loop system, which forms a Riesz basis with parenthesis for the state space. Then both the spectrum-determined growth condition and exponential stability...

On the economical solution method for a system of linear algebraic equations.

Awrejcewicz, Jan, Krysko, Vadim A., Krysko, Anton V. (2004)

Mathematical Problems in Engineering

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