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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 ...
Thomas Petit's user avatar
-2 votes
0 answers
20 views

regularity of the weak solution on the cube [closed]

Let $Q:=[0,1]^d$ and $g\in L^2(\Omega)$ consider the PDE : $$ \left\{ \begin{array}{ll} -\Delta f=g & \text{in $Q$} \\ f\equiv 0 & \mbox{on $\partial Q$}, \end{array} \...
Alucard-o Ming's user avatar
-3 votes
1 answer
31 views

Failure of Schauder estimates in $L^{\infty}$ [closed]

How should I go proving that (say) there are no estimates on the hessian of $u$ of the form $\|\nabla^2u\|_{\infty} \leq C \|f\|_{\infty}$? (Here one can consider $f$ continuous on the closure of the ...
Paul B's user avatar
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2 votes
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On fundamental solutions of parabolic PDEs

Let $U$ be an open bounded subset of $\mathbb{R}^n$ and consider the PDE in $U \times [0, T] \subset \mathbb{R}^n \times \mathbb{R}$, given by $$ \begin{cases} \partial_t \rho(x, t) = \frac{1}{\alpha} ...
mathusername's user avatar
1 vote
0 answers
26 views

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 $$\mathcal{L}v := -\Delta v - \lambda v.$$...
Santos's user avatar
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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 ...
RiaDoog's user avatar
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0 answers
40 views

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 ...
sam wolfe's user avatar
  • 3,405
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11 views

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 ...
AMHG's user avatar
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4 votes
2 answers
111 views

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 ...
Bananach's user avatar
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The modulus of $H^1_0(\Omega)$ function is also in $H^1_0(\Omega)$

Let $u \in H^1_0(\Omega)$. I want to prove that $|u| \in H^1_0(\Omega)$. The book I am reading gives a tip: Define $\zeta_\epsilon (t) = (t^2 + \epsilon^2)^{1/2} - \epsilon$ and show that $\zeta_\...
Lucas Linhares's user avatar
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1 answer
29 views

Convergence of modulus of a approximating sequence in $H^1_0(\Omega)$

Let $u \in H^1_0(\Omega)$ and consider $u_n \in C^\infty_0(\Omega)$ such that $\|u_n - u\|_{H^1(\Omega)} \to 0$. I know $|u| \in H^1(\Omega)$. I would like to know if $|u_n| \to |u|$ in $H^1(\Omega)$. ...
Lucas Linhares's user avatar
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1 answer
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Extension of Pohozaev's identity to the whole domain $\mathbb R^m$

TLDR, I am trying to extend Pohozaev's identity on bounded domains to the unbounded domain $\Omega=\mathbb R^m$ but I am having some difficulty. Would appreciate some help. I recently proved the ...
K.defaoite's user avatar
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Regularity of Kantorovich potentials for general cost function

I know De Philippis and Figalli have a paper studying the regularity of the Kantorovich potential. In Theorem 3.3, the authors show that the potential is $C^{k+2, \beta}$ if the density functions of ...
tianer555's user avatar
1 vote
0 answers
41 views

Time regularity vs space regularity for parabolic PDE

Suppose that there exist separable Hilbert spaces $V, H, X$ such that $V\hookrightarrow H\hookrightarrow X\hookrightarrow V'\,$ continuously, where $V'$ denotes the dual of the Hilbert space $V$. Let ...
Azam's user avatar
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1 vote
1 answer
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Whether functions in 2D Sobolev Space $W^{1,2}$ space are countinuous(a.e.)?

Q1:Whether functions in 2D $W^{1,2}$ space are continuous(a.e.)? I've learned that functions in 1D $W^{1,p}$ are a.e. absolute continuous.(Evans PDE chapter5 problem4) I also know that functions like $...
Cabbagebislikethis's user avatar
2 votes
1 answer
50 views

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 ...
Scottish Questions's user avatar
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0 answers
12 views

Concluding a radial weak solution with radial test functions is a weak solution with all test functions

Let $B \subset \mathbb{R}^N$ be the unit ball and $E = \{u \in H^1_0(B) : u \text{ is radial}\}$. Define the functional $I : E \to \mathbb{R}$ by $$ I(u) = \int_{B} |\nabla u(x)|^2 dx - \frac{1}{p}\...
Lucas Linhares's user avatar
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1 answer
52 views

Elliptic regularity theory and radial functions as test functions

Let $B$ the unit ball in $\mathbb{R}^N$ and $E = \overline{C^\infty_{0,r}(B)}^{H^1_0(\Omega)}$ the space of radial functions in $H^1_0(B)$, where $C^\infty_{0,r}(B) = \{ u \in C^\infty_0(B) : u \text{ ...
Lucas Linhares's user avatar
1 vote
1 answer
130 views

Regularity of weak solution of elliptic equation with nonlinear Neumann boundary

Let $\Omega \subset \mathbb{R}^n$ be bounded smooth domain and $u \in W^{1,2}(\Omega;\mathbb{R}^m)$ be a weak solution to the following equation \begin{align*} &\int_{\Omega} \nabla u \cdot \nabla ...
mnmn1993's user avatar
  • 395
1 vote
0 answers
36 views

Sobolev regularity for the Poisson problem with discontinuous coefficient

Let $u \in H^1(\Omega)$ be the weak solution of the Poisson problem: $$ \begin{cases} -\nabla \cdot (\mu \nabla u) = f & \text{in } \Omega \\ u = 0 & \text{in } \partial \Omega \end{cases} $$ ...
numeralist's user avatar
3 votes
1 answer
41 views

$L^{2}H_{0}^{1}$-estimate for solutions to the heat equation using the heat kernel

Let us consider the heat equation \begin{equation} \begin{cases} u_t=u_{xx} & \text{in $\mathbb{R}\times (0,T)$}, \\ u(\cdot,0)=u_0 & \text{on $\mathbb{R}$}, \\ \lim_{|x|\to \infty}u(x,t)=0 &...
Kai's user avatar
  • 131
1 vote
0 answers
94 views

Using Fredholm Alternative theorem to solve PDEs [closed]

I am trying to solve a coupled PDE system using the perturbation theorem, where parameter $\epsilon$ is small and all the variables are expanded in even powers of $\epsilon$. The set of equations and ...
mathlearner's user avatar
6 votes
2 answers
223 views

The Variational form of a biharmonic PDE

Suppose $\Omega \subset \mathbb{R}^d$ is a $C^{1,1}$ domain. Consider the biharmonic boundary value problem (BVP): $$ \begin{cases} \Delta^2 u = f \\ \nabla u \cdot \nu = g \\ u = u_D \end{cases} $$ ...
Mr. Proof's user avatar
  • 1,533
1 vote
2 answers
78 views

Maximum principle of strong solution of linear parabolic equation in $\mathbb{R}\times [0,T]$

Suppose that there is a strong solution $u(x,t)\in W^{2,1}_{2,loc}(\mathbb{R}\times [0,T])$ solving the linear parabolic equation $$-\partial_t u +\partial_x^2 u +b(x,t)\partial_x u+c(x,t) u =0\quad\...
mnmn1993's user avatar
  • 395
4 votes
0 answers
90 views

The constant in Schauder estimate of linear elliptic PDE

Assume that $u(x)$ is the classical solution solving $$a_{ij}(x)\partial_{ij}u(x)+b_i(x)\partial_iu(x)+c(x)u(x)=f(x)$$ on $\mathbb{R}^n$ for some smooth enough coefficients and uniformly elliptic $a_{...
mnmn1993's user avatar
  • 395
1 vote
0 answers
56 views

Elliptic regularity for Dirichlet problem

Let $\overline{M}=M \cup \partial M$ be a compact manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$. Let $P$ be an injective ...
user505117's user avatar
0 votes
0 answers
14 views

A question about regularity results in the linear Elliptic case which are given by Schauder theory

I've been reading Jost's lecture notes "Nonlinear Methods in Riemannian and Kählerian Geometry". In section 2.2 he gives a regular results about Elliptic and parabolic equations, but he ...
luyao's user avatar
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1 vote
1 answer
34 views

How do we show that on a parabolic Hölder space, the polynomial and Hölder seminorms are equivalent?

I am currently working through Krylov's Lectures on Elliptic and Parabolic Equations in Holder Spaces. One of the key points of chapter 8 is that the two seminorms $$[u]_{1+\delta/2,2+\delta;U} := \...
IdenticallyEulerian's user avatar
0 votes
0 answers
32 views

Regularity of weak solutions with $C^2$ boundary - Evans

I'm reading the proof of Theorem 4 from section 6.3 of Evans - Partial Differential Equations. I'm looking for an explanation about the following claim Previously, I already proved that $u_k\in H^1(V)...
matdlara's user avatar
  • 377
1 vote
0 answers
25 views

Integration by Parts for not so regular Sobolev functions

I am concerned with the following question: Let us assume we have some nice bounded domain $\Omega$ and $u\in W^{2,p}(\Omega)$ for some $1<p<2$. Let us further assume that we know that $-\Delta ...
micha's user avatar
  • 41
0 votes
0 answers
30 views

Higher regularity for linear parabolic equation with time depndent coefficient

I am looking for a higher regularity result for the solution of the problem $$\partial_t u+div(-A(t,x)\nabla u)=f$$ in a bounded smooth domain $\Omega$ with ...
user1288096's user avatar
2 votes
0 answers
170 views

$W^{1,p}(\Omega)$ estimates of solutions of elliptic equation

Consider the following simple elliptic problem in a bounded domain $\Omega\subset\mathbb{R}^N$: $$ -\Delta u(x) + u(x) = f(x) \qquad \forall x\in \Omega $$ with Neumann boundary conditions in $\...
joaquindt's user avatar
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0 answers
32 views

Regularizing kernel and mollifier

I am learning the book "Optimal transport old and new" by Villani. We have the following notion of Regularizing kernels. My first question is: if this regularizing kernel is similar to ...
Jay's user avatar
  • 301
1 vote
0 answers
51 views

Regularity of solution of heat equation

Let consider on $[0,T]\times\mathbb{R}$ the heat equation: \begin{equation} \tag{1} \label{heat} u_t = (\alpha u_x)_x\text{ with } u(0,x) = u_0(x)\in\mathcal{C}^\infty(\mathbb{R},\mathbb{R}). \end{...
NancyBoy's user avatar
  • 504
0 votes
0 answers
23 views

Wave equation on the half-line: is there always a $C^2$ solution?

This question has been asked again here but no answer was given. The problem is the following: \begin{align*} &u_{tt}=u_{xx}, \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ \mathbb{R}_+ \times (0,\infty) \\...
Plemath's user avatar
  • 449
0 votes
0 answers
76 views

vector field PDE sandwich theorem

Consider the vector valued system of PDE's for $\textbf{f} = [f_1,f_2,f_3]$ and $\textbf{g} = [g_1,g_2,g_3]$ \begin{equation} \dfrac{\partial}{\partial t}\textbf{f}- F\left(\textbf{f}, \nabla f_i, \...
MrPie 's user avatar
  • 235
1 vote
1 answer
46 views

When can we apply sandwich or blow-up criterion? [closed]

Suppose we have two PDE's like $$\dfrac{\partial}{\partial t} f - F(\dfrac{\partial f}{\partial x_1}, \dfrac{\partial f}{\partial x_2}, \dfrac{\partial f}{\partial x_3}, f) = 0 $$ and $$\dfrac{\...
MrPie 's user avatar
  • 235
1 vote
0 answers
40 views

A problem about elliptic differential equations

Suppose $u\in H_0^1(\Omega)$ is the weak solution of the following equations: $$ \begin{cases} -\left(a^{ij}(x)u_i\right)_j +b^iu_i=\sin u(x),x\in \Omega\\ u(x)=0,x\in \partial\Omega \end{cases} $$ ...
Jason's user avatar
  • 27
0 votes
2 answers
33 views

Estimate for the operator $A A_D^{-1}$

Let $O\subset \mathbb{R}^d$ be a bounded domain of the class $C^{1,1}$ (or $C^2$ for simplicity). Let the operator $A_D$ be formally given by the differential expression $A=-\mathrm{div} g(x)\nabla$ ...
Yulia Meshkova's user avatar
0 votes
1 answer
55 views

Showing $\inf_{u\in C^1_0(\Omega) } \frac{\int_\Omega|\nabla u|^2}{\int_\Omega| u|^2} >0$

Assume $\Omega$ is a bounded and regular domain of $\mathbb{R} ^n$. Show that $$\inf_{u\in C^1_0(\Omega) } \frac{\int_\Omega|\nabla u|^2}{\int_\Omega| u|^2} >0. $$ My attempt was that since those $...
unknown's user avatar
  • 391
4 votes
2 answers
82 views

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 ...
Turtle5's user avatar
  • 388
0 votes
0 answers
24 views

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) &&...
snape1234's user avatar
1 vote
1 answer
68 views

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....
ali's user avatar
  • 194
0 votes
0 answers
59 views

Energy method for linear advection

I found this wikipedia article about well-posedness of PDEs and the energy method. They consider the following problem the advection problem $$ u(t,x):[0,T]\times [0,1] \rightarrow \mathbb{R} , \quad ...
l'étudiant's user avatar
0 votes
0 answers
12 views

Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function

Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$: Given a function $u(x,t) \in L^2([0,T];H^2(\mathbb{R}))$, and $\alpha(t), \beta(t)$ which are injective and continuous, we have that $$v(x,t) = \alpha(t)u(...
ali's user avatar
  • 194
1 vote
0 answers
37 views

Regularity of "spatial Sobolev functions"

Consider a function $f\in C^{\infty}(\mathbb{R},H^{s}(\mathbb{R}^{d}))$, i.e. a function of two variables $f(t,x)$ such that for any fixed $t\in\mathbb{R}$ it holds that $f(t,\cdot)\in H^{s}(\mathbb{R}...
B.Hueber's user avatar
  • 2,876
0 votes
0 answers
26 views

If $\partial_t u_n -\Delta u_n = f_n$ , $\partial_t u -\Delta u =f$ and $f_n \to f$ in $L_t^p L_x^q$, do we have $u_n \to u$ in $W^{1,p}_t L^q_x$?

The question is summarized in the title. For each $n \in \mathbb{N}$, let us consider the Cauchy problem \begin{equation} \partial_t u_n -\Delta u_n = f_n \text{ on } [0,T] \times \mathbb{T}^4 \text{ ...
Keith's user avatar
  • 7,829
3 votes
0 answers
83 views

Obtain $L^\infty$ estimate of Poisson Equation only using eigenfunction expansion

Let $\Omega\subseteq \mathbb{R}^d$ a bounded domain with sufficient smooth boundary. We consider the Poisson Equation $$\left\{\begin{array} --\Delta u = f, & \Omega\\ u=0, &\partial\Omega\end{...
MikeMichael_maths's user avatar
0 votes
0 answers
43 views

The regularity of two functions and their convolution

$\quad$ I've heard that the regularity of convolution of two functions is more affected by the function that is more regular. As examples, here are some I know. if $f\in\mathcal L^1$ and $g\in\...
daidaitx's user avatar
  • 177
0 votes
0 answers
29 views

$-\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)...
micha's user avatar
  • 41

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