# Questions tagged [regularity]

regularity of solutions of PDEs.

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### Elliptic regularity theory in $\mathbb{R}^2$

I recently encountered two papers discussing elliptic PDEs and variational methods. The first paper claims that according to regularity theory, the solution to $-\Delta u = ug(u^2)$ in $\mathbb{R}^2$ ...
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### Modulus of Continuity, Heat Flow, and Derivative Estimates

Given $f : \mathbf{R}^d \to \mathbf{R}$, define $P_t f$ by \begin{align} (P_t f)(x) = \mathbf{E} \left[ f (x + \sqrt{t} G) \right], \end{align} where $G \sim \mathcal{N} (0, I_d)$ is a standard ...
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### Techniques to estimate PDE which are elliptic in some directions and degenerate in others

I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some ...
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### Smoothness of critical elliptic problem

I am convinced I have seen results along the lines of: if $u \ge 0$ is an $H_0^1(\Omega)$ solution of $$-\Delta u = u^{q-1}$$ in $\Omega$ with $u=0$ on $\partial \Omega$ (here $\Omega$ is a smooth ...
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### Parabolic theory for singular coefficients on bounded domains (Reference Request)

In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems. Is ...
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### On the derivatives of the solutions of the heat equations with Neumann boundary condition

Let $\Omega$ be a smooth domain with compact closure. More precisely, $\Omega$ is a $C^\infty$ domain. We write $\partial \Omega$ for the boundary of $\Omega$, and denote by $n$ the inward unit ...
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### Quotient of solutions of a semilinear Dirichlet problem is $L^\infty$

I posted this on MathStackExchange, but it hasn't even got 10 views, so probably it is better to post here. I hope it is not inappropriate. I am reading a paper of Brezis and Oswald about existence ...
68 views

Consider the incompressible Navier-Stokes equation on $(0,T)\times \mathbb{R}^3$ for fixed $T>0$. If the sequence of mollified initial vorticities $(\omega_0^{\nu})_{\nu}$ is uniformly bounded in $... • 173 2 votes 0 answers 128 views ### Smoothness of distance function induced by Finsler metric Consider$\mathbb R^N$endowed with a smooth Finsler metric$\phi:\mathbb R^N\times S^N\to (0,+\infty]$. The smoothness assumption are both on$\phi(x,\cdot)$(being at least$C^{2,1}$) and$\phi(\...
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Let $d$ be a positive integer with $d \ge 2$. We write $x=(x_1,\ldots,x_{d-1},x_d)=(\hat{x},x_d)$ for $x \in \mathbb{R}^d.$ The standard inner product and the Euclidean norm on $\mathbb{R}^d$ are ...