# Tagged Questions

Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

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### Maximum principle of the gradient of harmonic extension under weak regularity assumption

I have a question which is very likely to be trivial, but I'm stuck on it! Suppose $f \in W^{1, 2}(B_2(0))$ and $\|{\nabla f\|}_{L^{\infty}(B_2(0)\setminus B_1(0))} < \infty$. Consider then the ...
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### Density of restrictions of $p$-harmonic functions on a hypersurface

Let $\omega,\Omega\subset\mathbb R^n$, $n\geq2$, be bounded smooth domains so that $\bar\omega\subset\Omega$. Let $1<p<\infty$. Define the boundary space $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...
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### mixed Dirichlet Neumann regularity for an elliptic equation

Here is a problem which may be easy for some of you but not for me. Statement of the problem: Denote $\Omega := \{ (x,y) \in (0, \infty) \times (-\infty,\infty) \}$. Let $f \in L^2(\Omega)$ then by a ...
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I see this proof on http://www.math.uiowa.edu/~lwang/cccalderon.pdf and I couldn't understand what he did. If $||f||_{L^p(B_{4})} = \delta$ is small and the measure $|\{ x \in B_1; M(|D^2u|^2)>N_1^... 0answers 92 views ### Limit Toward Discontinuous Point of Dirichlet Boundary Value The question arises from a paper on Schwarz's domain decomposition method (click here). We consider a bounded domain in$\mathbb{R}^2$and a curve splits it into two, see the figure below. Now we ... 0answers 36 views ### Constructing a family of domains for application of method of continuity in optimal transportation Anyway can help me about this paper? http://arxiv.org/pdf/math/0601086v4.pdf I want to ask page 20, The author want to construct a family of subdomain for using method of continuity. But I can't ... 0answers 104 views ### Basic doubt in a free boundary problem for the Laplacian I am studying the following article : http://hal.archives-ouvertes.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf In this article the authors considers$K \subset \{ x \in R^n ; x_1 = 0 \}$a smooth, ... 0answers 158 views ### Bunimovich stadium bouncing ball http://terrytao.wordpress.com/2008/07/07/hassells-proof-of-scarring-for-the-bunimovich-stadium/ I cannot relate how the eigenfunctions normalizaed correspond to the probabality distribution of the ... 0answers 171 views ### Result like Brezis-Kato Lemma for Biharmonic equation. Is there any result for the 4th order elliptic pde,like we do have for 2nd order pde: Consider the equation,\$\Delta^2 u = a(x)u$and$a(x)\in L^{\frac{n}{4}}(\Omega)$, where$\Omega$is bounded. If$...
Consider the following nonlinear elliptic equation $$-\triangle u + u + u^3 = g, \quad x \in R^3.$$ If $g \in L^2(R^3)$, then the set $Q$ of solutions of above equation is bounded in $H^2(R^3)$, and ...