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

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

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### If $u \in H_0^1(\Omega) \cap L^p(\Omega)$ and $\Delta u \in L^p(\Omega)$ then $u \in W^{2, p}(\Omega)$

How to show the following? If $u \in H_0^1(\Omega) \cap L^p(\Omega)$, $\Delta u \in L^p(\Omega)$ then $u \in W^{2, p}(\Omega)$. This is part of the Brezis-Kato regularity argument as presented by ...
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### $H^1$-conforming approximation for elliptic PDE with discontinuous coefficient?

everyone Suppose I want to solve the diffusion equation $$-\nabla\cdot a \nabla u=f, \\u=0 \text{ on } \partial \Omega,$$ $f \in L^2(\Omega)$, $\partial \Omega$ is smooth. I use the standard node-...
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### Euristic and intuitive idea behind the theory of viscosity solutions

As the title suggests, I am kinda struggling to understand the basic idea behind viscosity solutions theory. The theorems are a lot different from what I saw in classical theory for PDEs (with Sobolev,...
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### Interior $H^2$ regularity (proof)

The problem is related to the proof of the interior $H^2$ regularity theorem from Evans's book Partial Differential Equations (sec 6.3). In the proof, I am having difficulty in understanding how the ...
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### Heat equation and convergence of approximate solution

We consider the equation \begin{equation} \partial_t u(t,x)+ v(t,x) \cdot \nabla u(t,x)= \kappa \Delta u(t,x) + F(t,x,u(t,x)) \ \mbox{ dans } \mathbb{R}_+ \times \mathbb{R}^d,....(1) \end{equation} ...
Higher Interior regularity of second order elliptic equation i am not underderstand how the inequality (37) comes from (29)-(32) and (36), and how $\tilde{f}\in L^2(W)$ (?) \$||\tilde{f}||_{L^2(W)}...