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5 votes
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Where to locate $0\in \Omega$ to get $u_{\varepsilon}(0)\neq 0$ where $\Delta u_{\varepsilon} + (\lambda-\varepsilon) u_{\varepsilon} = \frac{1}{|x|}$

Let $\Omega \subset \mathbb{R}^3$ a smooth bounded domain with $0\in \Omega$ and $u_\varepsilon(x)$ the solution to $$ \Delta u_\varepsilon + (\lambda-\varepsilon) u_\varepsilon = \frac{1}{|x|}\quad \...
Jonas's user avatar
  • 241
2 votes
0 answers
183 views

Regularity of elliptic partial differential equation with mixed Dirichlet-Robin boundary condition, to prove $u\in H^{2}(\Omega)$

I have posted this problem on Math Stackexchange but got no reply. When I deal with the wave equation with dynamical boundary condition, I am confused by the regularity of the following elliptic ...
monotone operator's user avatar
2 votes
0 answers
329 views

General solutions for HJB equations in a special case.

I am reading the book of Wendell Flemming in control theorem to learn the HJB equation Here is the setting that interests me: Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ functions such that $0\...
Steven's user avatar
  • 281
1 vote
0 answers
106 views

Solution to hyperbolic linear second order PDE

I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...
SebastianP's user avatar
1 vote
0 answers
43 views

Mixed boundary condition of parabolic equations

Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that $$ \partial\Omega=\partial\Omega_D\cup\partial\Omega_N, $$ where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...
Luis Yanka Annalisc's user avatar
1 vote
0 answers
258 views

Is this Stefan-type problem an open problem?

I am looking for a weak-formulation that would give me an existence and uniqueness of a solution of a Stefan-type problem. It is basically a 2-phase Stefan problem in 2D except that the free-boundary ...
Kanna Nakamura's user avatar
0 votes
0 answers
183 views

Has this form of the heat equation been solved for the radiation boundary condition

Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...
Michael Bernthal's user avatar