# Tagged Questions

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### some auxiliary results

I want to prove the following If $N\ge 3$ there exists a constant $c_0=c_0(\Omega)$ such that for all $\alpha\ge 1$ and $z\in H^1(\Omega)$ \begin{align} ...
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### weak solution of poisson equation

Consider the equation , with $u\in H^1$ $$\begin{cases}\Delta u = f & \text{in }\Omega\\ \displaystyle \frac{\partial u}{\partial \nu} = 0 & \text{in } \partial \Omega\end{cases}$$ where $\nu$ ...
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### Smoothing effect for weak solutions of heat equation

Let $u_0 \in L^2$ and $f \in L^2(0,T;H^{-1})$ and consider the solution $u \in L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$ of $$u_t - \Delta u = f$$ $$u(0) = u_0$$ with some BC (eg. zero Dirichlet). I am ...
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### Local Mollification of a $L^1$ function

Let $f$ be a $L^1(\Omega)$ function, where $\Omega\subseteq \mathbb{R}^2$ is a smooth and bounded domain. Then can we find a fuction $\phi\in W_0^{2,2}$ suct that $\phi>0$ in $\Omega$ and ...
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### Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
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### A priori estimates for functions in $C_0^\infty (\overline{\Omega})$.

Let $u\in C_0^\infty (\overline{\Omega})$, where $\Omega\subset \mathbb{R}^N$ is a bounded domain. Fix some $a\in \Omega$ and choose $r>0$ such that $\overline{\Omega}\subset B(a,r)$. Define ...
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### If $u \in L^2(0,T;H^1)$ with $u_t \in L^2(0,T;H^1)$ and $u$ and $u_t$ are bounded, is $u \in C([0,T];L^\infty$)?

Let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$ so $$\int_0^T u(t)\varphi'(t) = -\int_0^T u'(t)\varphi(t)$$ holds for all $\varphi \in C_c^\infty(0,T)$. Suppose we know ...
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### Characterization of weak solution

5 Nonlinear elliptic variational inequalities Preliminaries In order to explain the importance of elliptic variational inequalities, first consider the weak solution of the linear ...
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### Regularity of a Weak Solution

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
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Let's consider the domain $U=[-\pi,\pi]\times[-1,1]$. Assume that we have two functions $f\in H^2$ and $g\in H^{1/2}$. I wonder if the following bound is true: $$\|f g_{x_1}\|_{H^{-0.5}(U)}\leq ... 2answers 23 views ### Inner Product Representation of Functional on H^1_0 Let T\colon H^1_0(\mathbb{R})\to \mathbb{R}, \;T(f)=\int f'\phi \;dx, where \phi\in L^2 is fixed. By Hölder, |T(f)|\leq\|\phi\|_2\|f'\|_2\leq C \|f\|_{H^1_0}, i.e. T is continuous. Therefore, ... 1answer 33 views ###  \int_U |Du|^2\leq (\int_U u^2)^{1/2} (\int_U |D^2 u|^2 )^{1/2} for u\in H^1_0(U)\cap H^2(U) This is an exercise in Evans's book PDE. For u \in C_c^\infty(U), we have$$ \int_U |Du|^2\leq (\int_U u^2)^{1/2} (\int_U |D^2 u|^2 )^{1/2}$$by Holder inequality and ... 2answers 53 views ### f\in W^{1,p}((0,1)) is absolute continuous This is an exercise in Evans's PDE book f\in W^{1,p}((0,1)) is absolute continuous where 1\leq p < \infty  Try : By definition of sobolev space, f has weak derivative f' So$$\ast\ ...
Here in Wikipedia, it is said that in the one-dimensional case, it is enough to assume that the $(k-1)$-th derivative of the function $f$, is differentiable almost everywhere and is equal almost ...