Questions tagged [dirichlet-problem]
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4 questions
3 votes
0 answers
66 views
Comparison principle of the Monge–Ampère equation for discontinuous boundary data
Let $u$, $v$ be convex functions on an open and bounded set $\Omega$ such that their lower semi-continuous envelopes satisfy $u_*\le v_*$ on the boundary and their Monge‒Ampère measures satisfy $\...
3 votes
1 answer
405 views
Existence of a harmonic function on a bounded open set in euclidean space extending a continuous function on its boundary
Let $U \subset\mathbb{R}$$n$ be an arbitrary connected bounded open set, and let $f$ : $∂U$ $\rightarrow$ $\mathbb{R}$ be continuous, where $∂U$ denotes the set $\overline{U} - U$. Are there ...
- 3,555
0 votes
0 answers
205 views
Generalized Laplacian
I was wondering if any of you had ever encountered operators on $L^2(\mathbb{R}^2)$ of the form $$ \nabla \cdot (A(x)\nabla) $$ where $A(x)$ is some symmetric matrix field (viewed as $L^2(\mathbb{R}^{...
- 41
0 votes
1 answer
73 views
Is a continuum in the plane regular for the Dirichlet problem at all points?
As the title asks. Let me elaborate; suppose $\mathcal K$ is a continuum (compact, connected) set of $\mathbb C$ (with at least two points!). Let's say that $g(z;a)$ is the green's function of the ...
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