# 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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### Radius of balls in estimates for De Giorgi method

I wanted to ask a question regarding the radius of balls used to get the different estimates to establish both the jump from $L^2$ to $L^{\infty}$ and the Holder continuity later on in the proof of De ...
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### Regularity of an elliptical problem

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain with boundary of class $C^{\infty}$, with $N \geq 3$. Moreover, consider the uniformly elliptic operator $$\mathcal{L}v := -\Delta v - \lambda v.$$...
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### Exercise 7.6 of Robinson, Rodrigo, Sadowski: Smoothness of Navier-Stokes on Bounded Domains

My question is about Exercise 7.6 of the excellent book 'The Three-Dimensional Navier-Stokes Equations' by Robinson, Rodrigo and Sadowski. More generally, it is about higher regularity in space of ...
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### How to classify and solve this PDE?

Consider the following equation $$\partial_t f(t, x) = C_1 \partial_x f(t, x) - x g(t) f(t, x) + 2 g(t) \int_{x}^{\infty} f(t, y) dy$$ for $t>0$, $x\in\mathbb{R}$, some function $g(t)>0$ and ...
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### Approximation with bounded function

Let $\mathbb{D}$ be the unit disc, and let $B(o,r) \subset B(o,r')$ be two balls contained in $\mathbb{D}$. Assume that we have a $C^{\infty}$ function $f: \mathbb{D} \to [a,b]$ which has all its ...
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### Does smoothness of solution to parabolic equation require smoothness of coefficients?

I have a function that solves a parabolic partial differential equation $$\partial_tu - Lu = 0$$ with a linear second order uniformly elliptic-in-space differential operator $L$, whose coefficients ...
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### Density of similiar Sobolev space

Consider the space of functions defined as, $$D = \{f \in L^p(0,\infty): f \in AC_{loc}(0,\infty) \text{ and } xf'(x) \in L^p(0,\infty)\},$$ where $AC_{loc}(0,\infty)$ is the set of locally ...
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### Why does my solution to $\Delta u = \lambda u$ contradict the regularity theorem?

I am confused about the regularity theorem for Laplacian. It states that if we take a weak solutions of $\Delta u = \lambda u$, then $u$ must be a smooth function. But I cannot understand this ...
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### Energy estimate Neumann Problem with paramter

Let $I=(0,1)$. Suppose that I have the following problem \begin{align}\label{eq:NBVP_FT} \left\{\begin{array}{rclclcl} -(\partial_t^2 - \lambda^2) {v} (\lambda,t) & = & {f}(\lambda,t) &&...
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### If $u(.,t) \in H^2(\mathbb{R})$, then $u (.,t) \in C^1(\mathbb{R})$?

The statement is as such: $u \in L^2([0,T]; H^2(\mathbb{R}))$ for any $T>0$, thus $u (.,t) \in > C^1(\mathbb{R})$ a.e. in $t>0$. My guess was that this must be a result of Sobolev embedding....
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### $-\Delta u =f$ for $f\in L^p(\Omega)$ implies $\nabla u \in L^p(\Omega)$
I wonder if there is a quick&dirty proof of the following statement: Assume $\Omega \subset \mathbb{R}^n$ open and bounded with $C^1$-boundary and $p\geq 2$. The unique solution \$u\in H^1_0(\Omega)...