Articles of regularity theory of pdes

Laplacian as a Fredholm operator

Let $\Omega$ be a bounded smooth domain in $\mathbb R^n$. The Laplacian $\Delta$ acts on functions on $\Omega$. From elliptic regularity (I haven’t worked out all the details), we have that $$ \Delta : C^{2,\alpha} (\Omega) \to C^{0, \alpha}(\Omega),\ \ \alpha >0$$ and $$\Delta : W^{k,p}(\Omega) \to W^{k-2, p} (\Omega),\ \ k\ge 2, p \in […]

Spectrum of Laplacian on Half line. $\left [0, \infty \right)$

I would like to calculate the spectrum of Dirichlet and Neumann Laplacian of the domain $\left [0,\infty \right)$. To be precise, Define the Operator $T$ on $L^2\left[0,\infty\right)$ as $Tf=-f”$ and $D(T)=H^2\cap H_0^1$. And $S$ on $L^2\left[0,\infty\right)$ as $Sf=-f”$ and $D(T)=H^2 \cap \left \{f\in H^2 | f'(0)=0 \right \}$. Find out the Spectrum of T and […]

Conservation Laws: Difference and Reasonability Weak solutions, Integral solutions, distributional solutions

Suppose we have for $T\in\mathbb{R}_{>0}$ a conservation law of the following general form \begin{align} \dot{u}(t,x)+(a(t,x,u(t,x)))_{x}&=0 && (t,x)\in(0,T)\times\mathbb{R}\\ u(0,x)&=u_{0}(x) &&x\in\mathbb{R} \end{align} with $u_{0}$ and $a$ given and $a$ sufficiently smooth. We call it initial value problem (IVP). If one is interested in defining non-classical solutions, there are several different definitions of “weak” solutions: $u\in C([0,T];L^{1}_{\text{loc}}(\mathbb{R}))$ is […]

Functions Satisfying $u,\Delta u\in L^{1}(\mathbb{R}^{n})$

In this paper, the authors assert that …the domain of realization of the Laplacian in $L^{1}(\mathbb{R}^{n})$ is not contained in $W^{2,1}(\mathbb{R}^{n})$ if $n>1$. However, it is continuously embedded in $W^{1+\beta,1}(\mathbb{R}^{n})$ for each $\beta\in(0,1)$. Therefore, if $u$ and $\Delta u$ are in $L^{1}(\mathbb{R}^{n})$, then any first order derivative $D_{i}u$, $i=1,\ldots,n$, belongs to $W^{\beta,1}(\mathbb{R}^{n})$for each $\beta$, and […]

Proving the range of operator is closed

I have a hard time understand (2) of the Fredholm alternative in Evan’s Appendix. To prove the image of $I-K$ is closed, what result from functional analysis is used? I am lost in understand the first sentence of the proof, up to equation (4). Could anyone explain to me what result in functional analysis is […]

uniqueness heat equation

Consider the heat equation, $(1)$ $u_t=u_{xx}+f(x,t)$, $0<x<1$, $t>0$ $(2)$ $u(x,0)=\phi(x)$ $(3)$ $u(0,t)=g(t)$, $u(1,t)=h(t)$ When one wants to Show the uniqueness of solution of problem $(1)-(3)$, s/he can use so-called energy method or use maximum principle. My Question: What is the difference between these method? Is the class of solutions change for each method? Thanks in […]

Equivalence between norms in $H_0^1(\Omega)\cap H^2(\Omega)$.

Based in many questions and answers like [1, 2, 3 ] and a comment a good comment here [4]. I would like to know that the space $H=H_0^1(\Omega)\cap H^2(\Omega)$ can be equiped with this norm $$\tag{1}\|\cdot\|_H=||\Delta\cdot||_{L^2}+||\nabla\cdot||_{L^2}+||\cdot||_{L^2},$$ this is the $H^2$-norm. Or, is it equipped with this one $$\tag{2}\|\cdot\|_H=||\Delta\cdot||_{L^2}+||\cdot||_{L^2}.$$ Can I say that if $H$ is […]

cutoff function vs mollifiers

Q1. What are cutoff function? What are mollifiers? I cannot distinguish the two. Could anyone give some concrete/simple examples of cutoff functions? And how they differ from mollifiers I did check wiki (so please, I do not want wiki type answer) Q2. How to obtain compact support using a cutoff function? Could anyone give a […]

Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ it is well-known that we have inequality (Rellich inequality) $$ ‎\Lambda_N ‎\int_{\Omega}‎\frac{u^2}{|x|^4}\mathrm{d}x ‎\leq ‎\int_\Omega ‎|\Delta u|^2 \, ‎\mathrm{d}x‎ $$ where $‎\Lambda_N=(‎\frac{N^2(N-4)^2}{16})‎$ is optimal constant and also it is known that […]

Concerning the proof of regularity of the weak solution for the laplacian problem given in Brezis

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim brezis, and I am a bit confused about the proof. The theorem is stated as: Let $\Omega \subset \mathbb{R}^n$ be an open set of class $C^2$ with $\partial \Omega$ bounded. Let $f \in L^2(\Omega)$ and $u \in H_0^1(\Omega)$ satisfies the weak formulation […]