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Results tagged with elliptic-pde
Search options questions only
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user 33741
Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
1
vote
0
answers
52
views
some superharmonic function as a universal lower bound on Lipschitz domains
Question: for any bounded Lipschitz domain $\Omega\subset\mathbb{R}^d$, does there always exist a nonnegative function $\phi\in C^2(\Omega)$ such that
$\phi$ vanishes on $\partial\Omega$
the normal …
- 5,380
4
votes
0
answers
148
views
optimal regularity for elliptic pdes with $div(L^\infty)$ right-hand side (Hodge decompositi...
Question: In a smooth, bounded domain $\Omega\subset \mathbb R^d$, is it true
that solutions $\phi_f$ of
$$ \begin{cases}
-\Delta \phi_f=\operatorname{div}f & \mbox{in }\Omega\\ \phi_f = 0 & \mbox{o …
- 5,380
2
votes
0
answers
145
views
Minimizing some $H^{-1}$ functional over (a subset of) probability densities in $R^d$
Let me consider the following subset of probability measures in $R^d$
$$
\mathcal{K}_M=\left\{0\leq u(x)\in L^1(R^d):\quad \int u(x)dx =1,\,\int|x|^2u(x)dx\leq M,\,\int u(x)|\log u(x)|dx\leq M\right\} …
- 5,380
3
votes
0
answers
211
views
integration by parts on a Lipschitz domain as $\epsilon\to 0$
For a fixed, bounded, smooth domain $\Omega\subset \mathbb R^d$ and any $u\in W^{1,1}(\Omega)$ with trace $u|_{\partial\Omega}=g\in L^1(\partial\Omega)$ one can prove that
$$
\lim\limits_{\epsilon\to …
- 5,380
3
votes
0
answers
164
views
Asymptotic behaviour of principal eigenfunctions and large deviations
Dear Math Overflowers,
I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more detai …
- 5,380
4
votes
1
answer
467
views
Nonsmooth version of Hopf boundary point lemma
Let
$$
Lu=-a_{ij}(x)\partial_{ij}u+b_i(x)\partial_i u
$$
be a uniformly elliptic operator, with $A(x)=(a_{ij}(x))$ positive-definite.
Here I'm only considering smooth coefficients, and the domain $\Om …
- 5,380
6
votes
1
answer
683
views
Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$
The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to
$$
-\Delta u=f\hspace{3cm}(1)?
$$
I'm of co …
- 5,380
1
vote
0
answers
59
views
Discrete-to-continuum convergence of principal Fokker-Planck eigenvalues
I am looking for a reference justifying the following statement.
Let $L^n$ be any "reasonably consistent" finite-difference approximation of the Fokker-Planck operator in dimension $d=1$
$$
Lu=-\parti …
- 5,380