# Questions tagged [regularity]

regularity of solutions of PDEs.

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### Regularity of a weak solution to an elliptic PDE with mixed boundary condition

I have a question on the regularity of a weak solution to an elliptic PDE with mixed boundary condition. Let $\alpha \in (0,1]$ and let $D$ be a bounded $C^{1,\alpha}$-domain. Let $x \in \partial D$ ...
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### If $\frac{\partial f}{\partial t}(x,t)$ exists a.e and $\frac{\partial^2 f}{\partial t \,\partial x }$ is continuous, can we improve a.e existence?

The question is as in the title. Let $f(t,x) : [0,1]^2 \to \mathbb{R}$ be a function which is $C^\infty$ w.r.t $x$ for each fixed $t$ and whose derivatives w.r.t $x$ are all absolutely continuous w.r....
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### Boundary regularity for heat equation

Consider the heat equation $u_t - \Delta u=0$ with $u = u_0$ on $\partial B \times (0,T) \cup B \times \{t=0\}$. We consider weak solutions $u \in C^0(0,T;L^2(B)) \cap L^2(0,T;u_0 + W_0^{1,2}(B))$ ...
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### Are $c$-edge-colored clique removal lemmas known when $c>2$?

The following is a rephrasing of the Induced Graph Removal Lemma by Alon, Fischer, Krivelevich, Szegedy: For all $k>0$ and all $\epsilon > 0$, there is $\delta > 0$ such that the following ...
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### regularity of ground state of Schrodinger operator

I have a probably naive question on the regularity of ground state of the Schrodinger operator: $\Delta u - Vu = Eu$, where $\lim_{|x|\rightarrow\infty}V(x) = +\infty$ and $V\in C^2$, and $E$ is the ...
1 vote
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### Visualization of an oscillation lemma

How can one visualize Theorem 4.2 on page 31 of this paper by Seregin, Silvestre, Šverák and Zlatoš? On the other hand, I have a clear visualization of a related result about how oscillation decay ...
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### Gluing of two solutions to the same parabolic equation

Consider the domain $[0,1] \times [0,T]$ and the uniformly parabolic operator $L -\partial_t$ with smooth coefficient. Suppose I have $u_1(x,t) \in C^\infty([0,1] \times [0,T])$ solving \begin{...
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### What is the critical exponent for irregular function in the Sobolev scale?

When I first saw the definition of general Sobolev spaces with real exponent I immediately got interested in the following problem: pick several of your favourite irregular functions/distributions and ...
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### Regularity bound

For $\Delta f_g = g$, can we prove that $f_g \in L^\infty_{\text{per}}(\Omega)$ and \begin{align*} \|f_g\|_{L^\infty_{\rm per}} \le c \|g\|_{L^\infty_{\rm per}} \end{align*} where $c$ does not ... 105 views

### A question of the book "Regularity Theory for elliptic PDE"

In the book "Regularity Theory for elliptic PDE", written by Fernández-Real, page 67, $\tilde{u}_{k}$ converge to $\tilde{u}$ only in $C^1$ norm, but the result is that we can get a ...
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### Minimal regularity for domains in Green's formula

The Green formula is well-known for smooth bounded domains of $\mathbb R^d$. My question is: What is the minimal regularity known for domains where Green's formula still holds?
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