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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3
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0answers
46 views

$H^{1/2}$ regularity for the pressure in stationary Stokes problem in bounded domains

I am wondering for my research if, given the classical Stokes problem in a bounded domain $\Omega\subset \mathbb{R}^n$, $n=2,3$, \begin{align} - \Delta v + \nabla p &= f \text{ in }\Omega\\ \nabla ...
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22 views

The monotonicity or convexity of an ODE regard to its coefficient

I am trying get some ideas on how to prove that the solution of the following ODE is monotone or convex in the constant $k$: $$f(x)-x(f'(x))^2+k(x-x^2)f'(x)+(x-x^2)f''(x)=0$$ where $k\in(0,1)$ and the ...
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1answer
34 views

Linearisation of a Quasilinear Elliptic PDE

I have noticed that the word 'linearisation' can have different meanings in different places in the literature. For example, if one has a second-order quasilinear elliptic PDE, what would be the ...
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44 views

Apparent error in Jost's Partial Differential Equations

Theorem 13.1.1 in Jost's Partial Differential Equations asserts that if $f \in L^\infty(\Omega)$, with $\Omega$ a bounded open set in $\mathbb{R}^2$, then $$ u(x) = \int_\Omega \log |x-y| f(y)\ dy $$ ...
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33 views

Regularity of Newton potentials vis-a-vis weak solutions

It is well known that there exist counterexamples to the standard elliptic regularity theorem (sometimes called the "shift theorem") for second order differential equations in the case of ...
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51 views

$H^2$-regularity in space for linear parabolic equation

Consider the second order linear parabolic PDE: \begin{eqnarray*} \partial_{t}u + Lu &= f && \text{ in $\Omega \times (0,T]$},\\ u &= 0 && \text{ on $\partial \Omega \times [...
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1answer
77 views

$H^{1}$ or $H^{-1}$?

I am studying PDEs and Sobolev spaces. In some examples, the solution and/or initial data lies in the Sobolev spaces $H^{1}$ or $H^{-1}$. Sincerely, I do not understand the reason why some solutions ...
1
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1answer
28 views

Regularity solutions of an elliptic PDE on the unit square

Consider a $u$ that solves $$ \begin{cases} Lu &= f \qquad \text{ in } U,\\ u &= 0 \qquad \text{ on } \partial U, \end{cases} $$ with $U = (0,1)^n \subset \mathbb{R}^n$ the unit hypercube, and ...
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13 views

Schwartz fundamental solutions

Say I have some diffusion parabolic pde with smooth coefficients. Under uniform ellipticity, Ito shows there exists a $C^\infty_b$ fundamental solution $p(t,x,y)$ (away from $t=0$). If we weaken to ...
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0answers
23 views

Solve a coupled PDE in a rectangle

We consider a coupled PDE in a rectangle $\Omega=(-1,1)\times(-1,1)$. For the simplicity, we assume the functions are periodic in $x_{1}$ direction. \begin{equation*} \left\lbrace \begin{split} \nabla\...
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0answers
25 views

Requirements of the initial condition in heat equation for the existence and uniqueness of the exact solution.

I found that the initial condition $u(x,0)=\phi(x)$ need to satisfy atmost finite number of discontinuities for the existence and uniqueness of exact solution through Fourier transforms for the heat ...
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1answer
27 views

Prove for elliptic PDE, $c<0$ implies $\sup_{\Omega} |u| \leq \sup_{\partial\Omega}\lvert \phi \rvert+ \sup_{\Omega}\lvert \frac{f}{c} \rvert.$

I have a problem that might be related to Alexandroff Maximum Principle. But I don't know how to prove it: Assume $\Omega$ is a bounded domain. Let $u \in C^2(\Omega) \cap C^0(\overline{\Omega})$ ...
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17 views

Exterior sphere condition and Lipschitz Condition on the boundary for elliptic PDE

I am currently stuck on a problem. Here is the description: Assume $\Omega$ is a $C^1$, bounded domain that satisfies the exterior sphere condition: For every point $x_0 \in \partial \Omega$, there ...
3
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1answer
56 views

How to determine whether a solution of a transport PDE is classical?

I have to determine whether the solution of:$$u_y+2u_x=0 , x>0,y>0 ,\quad u(x,0)=x ,x\geq 0 ,\quad u(0,y)=y ,y\geq0$$ is a classical solution. I found that the solution is:$$u(x,y)=\frac{-x+2y}{...
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0answers
8 views

Mollifiers for a function on $[0,T]\times R$

Do you have any precise and comprehensive reference for how to build a sequence $\phi^\epsilon(t,x)$ of $C^\infty([0,T]\times R)$ functions that converge uniformly on $(0,T)\times R$ to a given ...
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0answers
14 views

difference quotient methods in L1

Usually for some pde, if we want to prove that a given solution has higher derivatives, we proceed by difference quotient method. For elliptic pdes, the result is usually that some derivative belongs ...
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0answers
25 views

L^1 Gradient bounds for potentials of weakly closed forms.

The Poincare-lemma is a central statement in differential geometry. It shows that a k-form is closed iff it is exact. A special case is as follows: Let $\omega\in\Omega^k(U)$ with $\omega=\sum_{I\in\...
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1answer
99 views

A regularity result for semilinear PDE of the form $\Delta u=f(x, u)$ in Michael E. Taylor's book Partial Differential Equations III.

Under the assumption that $$\partial_{u} f(x, u) = 0 \text{ for } |u| \geq K \quad(1.6)$$ Michael E. Taylor said that (proposition $(1.3)$) For $k=1,2,..$, if $g \in H^{k+1 / 2}(\partial M)$ then any ...
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21 views

Regularity of $\partial_t u-\Delta u=f(u)$ via bootstrapping

The following question is motivated by this previous post after some things have been clarified. So, let us assume the problem $\begin{align} \partial_{t} u-\Delta_M u&=f(u) \quad \text{ in } M_T \...
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1answer
48 views

Regularity of Parabolic pde (via Boostrap argument?) and references needed

Let $M$ be a compact $2d$-manifold in $\mathbb R^3$ and take $T>0\;$. We set $M_T:=M \times (0,T)$ and we consider the following parabolic PDE \begin{align} \partial t u-\Delta_M u&=f \quad \...
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0answers
33 views

Regularity for a semilinear problem

we are facing the following problem: Let $\Omega \subset \mathbb{R}^d$ open bounded. Consider the elliptic operator $L = -div(a\cdot \nabla u)$, where $a_{i,j}$, where $a_{i,j}\in C^2(\Omega)$ are ...
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0answers
52 views

Partial Differential Equation with weird behaviour

I came across the following PDE: $$ \partial_xf(x,y)+\partial_y(f(x,y))=\gamma(x)(f^2(x,y)-f(0,y)) $$ Assume that $f$ is a well behaved function and $\gamma(x)$ is given and I have the initial ...
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1answer
31 views

Problem about multiplication of functions in Sobolev spaces and regularity related to gauge theory

Let $X$ be a closed Riemannian 4-manifold. Let $f\in L^2_k(X)$ (Sobolev space of maps $X\to \mathbb{C}$ of regularity $k$), with $k>2$. Suppose also that $f|_{\mathcal O} \not\equiv 0$ for any ...
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0answers
16 views

References regularity of solutions to linear system of first order PDE

Consider a linear first order PDE: $$\sum_i A_i(x)\frac \partial{\partial x_i}f(x) = B(x) f(x)$$ $f:\Omega\subset\mathbb{R}^n\to \mathbb{R}^m$ where $\Omega$ is bounded and $A_i,B :\Omega\to Hom(\...
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0answers
26 views

Hölder regularity of elliptic PDE

I am reading Gilbarg’s PDE. And I don’t know how to show the highlighted part in the proof of Theorem 6.17. I think Arzela-Ascoli lemma can only imply subsequential convergence. And I also don’t ...
2
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0answers
72 views

Integral version of the lebesgue differentation theorem

Given $f\in L^p(B_1)$, where $B_1$ is the unit ball centered at $0$ in $\mathbb{R}^d$, we know that the point $a\in B$ such that $$ \lim_{r\to 0} \frac{1}{|B(a,r)|} \int_{B(a,r)} |f-(f)_{B(a,r)}|^p =0 ...
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36 views

Regularity for the critical case $p=2^*$

Suppose i have a weak solution $u \in H^1_0(\Omega)$ of the problem \begin{cases} -\Delta u = |u|^{2^*-2}u, & x \in \Omega \\ u=0, & x \in \partial \Omega \end{cases} And $\...
3
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1answer
42 views

If $\int_{\Omega} [u_t + (f(u))_x ] \phi\, dt \,dx =0 $ for all $ \phi \in C_0^\infty$ ,then it's true for all $ \phi \in C_0$

Prove that: If $$ \DeclareMathOperator{\Dm}{\operatorname{d\!}} \int\limits_{\Omega} [u_t + (f(u))_x ] \phi \Dm t \Dm x =0$$ for all $ \phi \in C_0^\infty(\Omega) $ , then it holds even for ...
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0answers
21 views

One question about regularity theory

I will for example that describe the problem: Let $f\in L^2(0,1)$ \begin{align}(1)\quad \begin{cases}-u''(x)=f(x),\forall x\in (0,1)\\ u(0)=u(1)=0\end{cases}\end{align} Then $\forall \phi \in C^\infty ...
2
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0answers
122 views

Gradient estimate of degenerate parabolic equation

Suppose $u : [0,1] \times [0,T] \to \mathbb{R}$, we consider the problem \begin{equation} \left\{\hspace{5pt}\begin{aligned} &\dfrac{\partial u}{\partial t} - a(x,t)x^2\partial_x^2 u - b(x,t)x\...
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0answers
29 views

Parabolic equation with domain depending on time

Let $d_1(t)$ and $d_2(t)$ be smooth functions from $[0,T]$ to $\mathbb{R}$. Suppose $L$ is a uniform elliptic operator and $u :\mathbb{R} \to \mathbb{R}$, we consider the problem \begin{equation} \...
1
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1answer
103 views

elliptic PDE with discontinuous coefficients

Consider $ \nabla \cdot (\sigma \nabla u)=0 $ in the unit ball $B\subset\mathbb{R}^3$ with some nice boundary conditions. Further assume that $\sigma\in (L^\infty)^{3\times3 } $ is symmetric positive ...
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2answers
58 views

Estimate of supremum of a function by an integral

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, $n\geq 2$. For $r>0$, denote by $B_r(x_0)=\{x\in\mathbb{R}^n:|x-x_0|<r\}$ whose closure is a proper subset of $\Omega$. Let $u\in W^{1,p}(\...
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0answers
20 views

Nonlocal Harnack inequality

Can someone, please help me to understand on how to obtain the last estimate on page 19 of the following paper, which says follows by using Lemma 2.7 there. My main problem in understanding how the ...
1
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1answer
36 views

Gradient of fundamental solution of an elliptic PDE is also a solution far from the pole

Let $A(x)$ be a uniformly elliptic (positive definite) matrix with Holder coefficients defined on $\mathbb R^n$ ($n \geq 3$) and consider the elliptic PDE $$ \operatorname{div}(A(x)\nabla u(x))=0, ~x \...
2
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1answer
74 views

What does the phrase "standard elliptic estimates" mean?

Recently I have seen this phrase "standard elliptic estimates" or "elliptic regularity theory" (I guess they mean the same thing) for many times but I can't find its explanation on ...
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0answers
53 views

Regularity of solutions of heat eqaution with boundary conditions

Fix $T>0$. Let us consider a heat equation $\rho(t,x)$ with initial condition $\rho_0(\cdot):[0,1]\to\mathbb R$ on the interval $[0,1]$. Namely, $$\partial_t\rho(t,x)=\Delta \rho(t,x)$$ for every $...
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0answers
53 views

Is Poisson formula valid for the weak solution of Laplacian?

In the book "Regularity Theory for Elliptic PDE", here is a theorem as follows Theorem(Harnack's inequality). Assume $ u\in H^1(B_1) $ is a non-negative, is the weak solution for the ...
2
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1answer
74 views

Regularity of higher order elliptic problem on compact smooth manifolds with boundary

I have trouble in finding a source in the literature for the following result: Let $\overline{M}$ be a compact smooth manifold of dimension $n \in \mathbb{N}$ with interior $M$ and non-empty boundary $...
2
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1answer
65 views

Does this function live in $H^{\frac{1}{2}}(\partial \Omega)$

In the Poisson equation $$-\Delta u=f$$ $$u=g \in \partial \Omega$$ we usually see in textbooks the requirement $g \in H^{\frac{1}{2}}(\partial \Omega)$ and $u \in H^1(\Omega)$. In my case, $\Omega = [...
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0answers
67 views

Existence of a weak solution to elliptic PDE with nonsmooth Robin boundary condition

Let $\Omega\subset \mathbb{R}^N$ with a smooth boundary $\partial\Omega$ and consider the elliptic PDE \begin{alignat*}{4} \mbox{D.E.} & \quad{\displaystyle \Delta u -u =0 },&& \quad x\...
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1answer
44 views

Is $(-y,x)/\sqrt{x^2+y^2}$ in $[W^{1,\infty}([0,1]^2)]^2$

in an application I have the following vector field $$\beta(x,y)=\frac{1}{\sqrt{x^2+y^2}}(-y,x)$$ and I need to check if this lives in the Sobolev space $[W^{1,\infty}([0,1]^2)]^2$ Looking at the ...
1
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1answer
38 views

Vector spaces in PDE $C^{1}([0,+\infty) ; H) \cap C([0,+\infty) ; D(A))$

What is the meaning of the following statement: $$ u \in C^{1}([0,+\infty) ; H) \cap C([0,+\infty) ; D(A)) $$ I am confused because I know that $C^{1}$ functions are continuous. I faced these spaces ...
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0answers
19 views

Getting Weak Harnack inequality with optimal constant

I am trying to find the following proofs: Given a solution to a linear elliptic equation $Lu=0$ with ellipticity bounds $\lambda, \Lambda$, we know that the solution is Holder continuous with exponent ...
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0answers
18 views

Unbounded initial data: Which solution space?

Suppose we are given an initial value problem $$ u_t=u_{xx} +f(u), \quad x\in \mathbb{R}, t>0, $$ with an initial datum $u(x,0)$ which is unbounded. Does it then make sense to search for (...
2
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1answer
64 views

Why do we write $H_0^1(\Omega) \cap H^2$ instead of only $H^2_0(\Omega)$?

I've seen that when we deal with Poisson equation with homogeneous boundary conditions, let's say in 2D with a convex domain $\Omega$, we write that the regularity of $u$ is $H^2 \cap H_0^1(\Omega)$. ...
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0answers
20 views

Explicit bound for $C^{2,\alpha}$ elliptic theory

Let $\Omega$ be an open, bounded with $C^2$ boundary (or smooth as we want). A result about elliptic regularity is given as follows. If $\Omega_0\subset\subset \Omega$ and $u$ is a weak solution of $...
3
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0answers
49 views

About EXISTENCE for PDE

In doing optimal control of Parabolic PDE's we often have to solve a problem like this: $$\begin{cases} \dfrac{\partial y}{\partial t}-d\Delta y(t,x)=f(y(t,x),p(t,x)) & (t,x)\in (0,T)\times\Omega ...
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0answers
42 views

Solution to the linear multiplicative heat equation via iteration in the mild formulation.

Consider the linear multiplicative heat equation on $\mathbb{R}^+\times \mathbb{R}^d$ given by: \begin{equation}\label{linear multiplicative heat equation} \partial_t u= \frac{1}{2}\Delta u + \xi ...
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0answers
34 views

Embeddings into $C^0([0, T]; X)$

Let $\Omega \subset \mathbb R^n$, $n \in \mathbb N$, be a smooth bounded domain, $T > 0$ and $p, q \in (1, \infty)$. (If it simplifies matters, I am also fine with just considering the special case ...

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