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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$H^{1/2}$ 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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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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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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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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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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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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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 ...
36 views

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 $$\operatorname{div}(A(x)\nabla u(x))=0, ~x \... 1answer 74 views What does the phrase "standard elliptic estimates" mean? Recently I have seen this phrase "standard elliptic estimates" or "elliptic regularity theory" (I guess they mean the same thing) for many times but I can't find its explanation on ... 0answers 53 views Regularity of solutions of heat eqaution with boundary conditions Fix T>0. Let us consider a heat equation \rho(t,x) with initial condition \rho_0(\cdot):[0,1]\to\mathbb R on the interval [0,1]. Namely,$$\partial_t\rho(t,x)=\Delta \rho(t,x)$$for every ... 0answers 53 views Is Poisson formula valid for the weak solution of Laplacian? In the book "Regularity Theory for Elliptic PDE", here is a theorem as follows Theorem(Harnack's inequality). Assume  u\in H^1(B_1)  is a non-negative, is the weak solution for the ... 1answer 74 views Regularity of higher order elliptic problem on compact smooth manifolds with boundary I have trouble in finding a source in the literature for the following result: Let \overline{M} be a compact smooth manifold of dimension n \in \mathbb{N} with interior M and non-empty boundary ... 1answer 65 views Does this function live in H^{\frac{1}{2}}(\partial \Omega) In the Poisson equation$$-\Delta u=fu=g \in \partial \Omegawe usually see in textbooks the requirement g \in H^{\frac{1}{2}}(\partial \Omega) and u \in H^1(\Omega). In my case, \Omega = [... 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\Delta u -u =0 },&& \quad x\... 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)$$and I need to check if this lives in the Sobolev space [W^{1,\infty}([0,1]^2)]^2 Looking at the ... 1answer 38 views Vector spaces in PDE C^{1}([0,+\infty) ; H) \cap C([0,+\infty) ; D(A)) What is the meaning of the following statement:$$ u \in C^{1}([0,+\infty) ; H) \cap C([0,+\infty) ; D(A)) $$I am confused because I know that C^{1} functions are continuous. I faced these spaces ... 0answers 19 views Getting Weak Harnack inequality with optimal constant I am trying to find the following proofs: Given a solution to a linear elliptic equation Lu=0 with ellipticity bounds \lambda, \Lambda, we know that the solution is Holder continuous with exponent ... 0answers 18 views Unbounded initial data: Which solution space? Suppose we are given an initial value problem$$ u_t=u_{xx} +f(u), \quad x\in \mathbb{R}, t>0, $$with an initial datum u(x,0) which is unbounded. Does it then make sense to search for (... 1answer 64 views Why do we write H_0^1(\Omega) \cap H^2 instead of only H^2_0(\Omega)? I've seen that when we deal with Poisson equation with homogeneous boundary conditions, let's say in 2D with a convex domain \Omega, we write that the regularity of u is H^2 \cap H_0^1(\Omega). ... 0answers 20 views Explicit bound for C^{2,\alpha} elliptic theory Let \Omega be an open, bounded with C^2 boundary (or smooth as we want). A result about elliptic regularity is given as follows. If \Omega_0\subset\subset \Omega and u is a weak solution of ... 0answers 49 views About EXISTENCE for PDE In doing optimal control of Parabolic PDE's we often have to solve a problem like this:$$\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 ...
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 ...
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 ...