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

This tag is for questions concerning the smoothness of weak solutions to partial differential equations.

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### Regularity for heat equation with Neumann boundary conditions

Consider the following inhomogeneous heat equation for $f=f(x,t)$ with Neumann boundary conditions on a bounded domain $\Omega$ and initial value $f(x,0)=f_0(x)$: \begin{align*} \partial_t f - \Delta ...
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### De Giorgi-Nash-Moser estimattes for non-symmetric a_ij

Is there any references on De Giorgi-Nash-Moser estimates for the parabolic operator $$\partial_t - \partial_j[a_{ij}(t, x)\partial_i]$$ where $a_{ij}$ is not symmetric?
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### Bound for $\left\|\nabla^2 u\right\|_{L^2(\Omega)}$

Consider the elliptic equation \begin{aligned} -\nabla \cdot A \nabla u=f, & \text { in } \Omega, \\ u=0, & \text { on } \partial \Omega. \end{aligned} where $\Omega$ is a bounded domain....
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### Heat semigroup in Heisenberg groups properties

I'm trying to find the construction of the heat semigroup in Heisenberg groups, but I haven't found anything. In particular, I'm trying to find out if the semigroup in the Heisenberg group has the ...
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### Parabolic regularity on a $C^\infty$ compact manifold

I have a general parabolic PDE on a compact $C^\infty$ manifold $M$ $\textit{ie}$ a (the?) solution $u$ is a probability density and satisfies the PDE in a weak sense (with $C^\infty_c$ test functions)...
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### Proving that a $W^{1,2}$ type space is Hilbert

Split the $N-$dimensional Euclidian space as $\mathbb{R}^N = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$. A vector in $\mathbb{R}^N$ will be denoted by $z = (x,y)$. Let $\alpha > 0$ and consider the ...
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### Caffarelli-Silvestre Extension problem (Existence of Solution of Degenerate elliptic PDEs in a ball)

In the Caffarelli-Silvestre paper link of paper before Lemma 4.2, it is given that \begin{align} \text{div}(|y|^a \nabla u)=0 ~~~\text{in the weak sense in} ~B_R\\ u=g \hspace{3.3cm}\text{on}~\partial ...
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### About $\epsilon$-Regularity theorem in Leon Simon's Theorems on Regularity and Singularity of Energy Minimizing Maps

I am reading Theorems on Regularity and Singularity of Energy Minimizing Maps by Leon Simon. The problem arises in Section 2.3, the proof of Shoen-Uhlenbeck theorem: Let $u:\Omega \to N$ is a ...
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### References for regularity of solution to fourth order elliptic/parabolic PDEs

After searching, all I could find was regularity of solutions to second-order PDEs like in Partial Differential Equations by Evans and Functional Analysis, Sobolev Spaces and Partial Differential ...
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### Finite time blow-up for solutions of a quasi-parabolic system

I am reading a book about the Harmonic Map flow and in the proof of the existence of solution the author states something I am not totally familiar with. Let me recall the statement of the theorem. ...
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### Regularity on the Poisson equation on bounded domains

This is a follow-up question to Regularity of solution for Poisson equation on bounded domains. We consider the Poisson equation $-\Delta u=f$ in $\Omega$ and $u=\varphi$ on $\partial\Omega$. ...
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### Can we prove elliptic regularity by reducing to a first order system of PDE?

Theorem 10.3.6 of Nicolaescu's notes prove elliptic regularity for a first order PDO. I noted that this PDO is allowed to be interpreted as a system of first order pde's. Now, given any linear, ...
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### Regularity of heat equation with nonhomogeneous boundary conditions

Consider the heat equation \begin{equation*} \begin{cases} u_{t} - \Delta u = f &\text{in }(0,T)\times\Omega\\ u(x,t) = g(x,t) &(t,x)\in [0,T]\times\partial\Omega\\ u(x,0) = h(x) &\text{on ...