# Questions tagged [regularity-theory-of-pdes]

The concept of regularity concerns the smoothness of weak solutions to partial differential equations.

38 questions
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### Estimate on average with weight

Let $B_{2R}=B(x,2R)$ be the ball of radius $2R$ centered at $x$ and $v=log\,u$ for some positive function $u$ defined on $B_{2R}$. Denote by $v_{B_{2R}}=\frac{1}{w(B_{2R})}\int_{B_{2R}}v(x)w(x)\,dx$, ...
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### Is a Sobolev map with smooth minors smooth?

$\newcommand{\Cof}{\text{cof}}$ Let $d>2$. Let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$ where $\Omega$ is an open subset of $\mathbb{R}^d$. Let $2 \le k \le d-1$ be fixed. Suppose that $\det df>0$...
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### Interior $H^2$ regularity - applying “Cauchy's inequality with $\epsilon$”

This is from PDE Evans, 2nd edition, pages 327, 328, and 330. I have a question regarding one piece of the proof. The theorem concerned is THEOREM 1 (Interior $H^2$ regularity) which is stated on ...
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### Proving weak coercivity by young's and interpolation inequalities

Let be $$(P)\left\{\begin{array}{ll} &-\Delta u + V(x)u=f & \text{ in }\ \Omega\\ &u=0 & \text{ on } \ \Gamma \end{array}\right.$$ with $V \in L^r(\Omega)$, for some $1<r<\infty$....
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### $\Delta u = f , \operatorname{div} f=0 \Rightarrow \operatorname{div}u=0$ on non convex domain.

I am specifically referring to this paper and why equation (7.7) is the weak formulation of (7.6). My question is why $\operatorname{div} u=0$ is implied by formulation (7.7) on a general domain. If ...
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### Unicity of solution for a parabolic problem?

How can I show that the parabolic problem $$\begin{cases} \partial_xu- \Delta u=0 & \mathbb{R} \times (0,+ \infty)\\ u(x,0)=f(x) & \mathbb{R} \end{cases}$$ has a unique solution? Can I ...
### Positivity of a normal derivative of $C^2$ function
Let $\Omega$ be an open set of $R^n$ of classe $C^2$, let $f \in {C^2}({\bar \Omega } )$ which satisfies the following properties: $f>0$ on $\Omega$ and $|\nabla f|>0$ on $\bar \Omega$. and \$f=...