2
votes
1answer
128 views

Strong maximum principle for weak solutions

Suppose I have a linear parabolic equation with solutions in the Bochner-Sobolev spaces (eg. $L^2(0,T;H^1) \cap H^1(0,T;H^{-1})$). Is it possible to obtain a strong maximum principle with a proof that ...
0
votes
1answer
133 views

Weak solution of a heat equation is zero?

I work on a bounded domain in $\mathbb{R}^n$. Let $u \in H^1(0,T;H^{-1})\cap L^2(0,T;H^1)$ be a solution of the heat equation: $$\langle u', v \rangle + \int \nabla u \nabla v = 0$$ for each test ...
1
vote
0answers
97 views

Compact embedding

Let $\Omega$ be a domain in $\mathbb{R}^d$ (not necessarily bounded, no regularity assumption) and $K \subset \Omega$ a compact. Is it true that the embedding $H^1_0(\Omega) \rightarrow ...
1
vote
1answer
128 views

Weak convergence of a sequence

I have a sequence $(u_k) \in L^2_{loc}(\mathbb{R}^+; H^1_0(\Omega) )$ and $u \in L^2_{loc}(\mathbb{R}^+\times \Omega )$ such that for any $T >0$ and any compact $K \subset \Omega$ we have : ...
2
votes
0answers
51 views

Approximating a superharmonic function, by smooth superharmonic functions

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$, $u\ge 0$ and $-\Delta u\ge 0$ in the sense of distributions ($u$ is superharmonic). The standard ...
0
votes
0answers
127 views

Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$

We have a bounded domain $\Omega$ of $\mathbb{R}^n$. Let $$u \in L^2((0,T);H^1(\Omega)) \cap H^1((0,T);H^{-1}(\Omega))\cap L^\infty((0,T);L^\infty(\Omega)).$$ I want to show for $r \geq 2$ that ...
0
votes
1answer
164 views

A question about PDE argument involving monotone convergence theorem and Sobolev space

I'm reading this paper. In it there is 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.. ...
1
vote
0answers
69 views

$L^p$ norm of solution to porous medium equation decreases in time: how to make formal calculation rigorous?

Let $u \in C^0([0,\infty);L^1(M)) \cap W^{1,1}_{\text{loc}}((0,\infty);L^1(M))$ with $u(t) \in H^1(M)$ for a.e. $t$ be the solution of the porous medium equation $\dot u = \Delta (u^m)$ on a compact ...
0
votes
1answer
135 views

Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary [on hold]

I'm interested in degenerate parabolic equations posed on compact manifolds without boundaries and in particular decay estimates of the weak solution of such equations of the form $$|u(t)|_{L^p} \leq ...
0
votes
0answers
128 views

$b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{-1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$

This question stems from the proof of Theorem A.1 on page 425 of this paper. Let $Q=(0,T)\times \Omega$. Suppose $b_n \rightharpoonup b$ in $L^q(Q)$ for any $q < \infty$ and $b_n \to b$ in ...
1
vote
1answer
140 views

Getting existence for $L^1$ data given existence for $L^\infty$ data and $L^1$ continuous dependence result

Let $F:\mathbb{R} \to \mathbb{R}$ be locally Lipschitz, monotone and continuous. For the sake of concreteness only let us suppose it is of porous medium type (eg. $F(r) = r^{\frac 1m}$.) Let $\Omega ...
0
votes
1answer
109 views

Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$. Define the truncation function$$T_\epsilon(x) = \begin{cases} ...
1
vote
0answers
148 views

Is this function space a “classical” Sobolev space?

I apologise if this is indeed classical but my functional analysis is quite rusty... My work recently led me to the norm: $(\|u\|_p)^p=\int_D (|u|^p+|\Delta u|^p)d\lambda$ where $D$ is the unit disk ...
1
vote
1answer
88 views

Getting a comparison principle for parabolic equation when solution is not that smooth

Consider the solution $b(u) \in L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$ with $u \in L^2(0,T;H^1)$ to $$\frac{\partial}{\partial t}b(u) - \Delta u = f$$ where $b$ is continuous, increasing and locally ...
5
votes
1answer
182 views

Chain rule for distributional derivative

Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$). Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...
2
votes
1answer
72 views

A bound in Sobolev spaces of negative order

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 ...
2
votes
1answer
99 views

Existence of solution with $u' \in L^2(0,T;L^2)$ to a nonlinear parabolic PDE

Consider the problem of finding $u \in L^2(0,T;H^1)$ with $u' \in L^2(0,T;L^2)$ such that $$\int_0^T \int_{\Omega}u'(t)\varphi(t) + \int_0^T \int_{\Omega}\nabla (F(u(t)))\nabla \varphi(t) = \int_0^T ...
3
votes
2answers
187 views

Negative real order Sobolev spaces: density and representation

First, I give my motivation to ask this question. The generalised Neumann trace can be defined as $$ {}_{H^{-1/2}(\partial\Omega)}\langle\frac{\partial ...
1
vote
0answers
95 views

Showing a normal-derivative operator is a (sort of) contraction (related to Crandall-Liggett and PDEs)

Denote by $\mathbb{E}(g)$ the solution of the PDE $$\Delta v(x,y) = 0 \quad\text{in $\Omega$}$$ $$v(x,0) = g(x) \quad\text{on $\partial\Omega$}.$$ Let $X=L^1(\partial\Omega)$. Let $h(t)$ be a ...
5
votes
0answers
107 views

A question on density of Lipschitz functions in weighted Sobolev spaces

Recall that for a domain $\Omega\subset \mathbb{R}^n$, the weighted Sobolev space $W^{1,n}(\Omega,\mu)$ is defined as $f\in L^n(\Omega,\mu)$ and the weak derivative $Df\in L^n(\Omega,\mu)$. Let now ...
3
votes
1answer
74 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
2
votes
1answer
192 views

A comparison principle for parabolic equation

(Crossposted from http://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear) Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for ...
1
vote
1answer
111 views

Getting an a priori bound on a nonlinear gradient term in PDE; how to adapt trick from $L^2$ case to $H^{-1}$ case?

I have the PDE $$u_t(t) - \Delta f(u(t)) = 0$$ in $H^{-1}(\Omega)$ where $f$ is a nonlinear function. Define $F(s) = \int_0^s f(s)$. Note that if $u_t(t) \in L^2(\Omega)$, $$\frac{d}{dt}F(u(t)) = ...
6
votes
0answers
114 views

Reference Request: Elliptic differential operators in the Fréchet setting

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...
0
votes
1answer
52 views

properties of frequency-uniform decomposition operator $\square_{k}^{\sigma}$

Let $\rho \in S(\mathbb R^{n})= \text{Schwartz space}, \ \rho:\mathbb R^{n}\to [0,1]$ be smooth radial function verifying $\rho(\xi)=1$ for $|\xi|_{\infty}\leq \frac{1}{2}$ and $\rho(\xi)=0$ for ...
3
votes
1answer
110 views

Techniques to show existence for a PDE with dynamic boundary condition

Let $\Omega$ be a bounded domain. I am looking for techniques to show existence of solutions to dynamic boundary problems of the form $$\Delta u = 0 \quad\text{on}\quad \Omega \times (0,T)\\ ...
2
votes
0answers
62 views

Changing the test function space in a weak formulation of parabolic PDE

Suppose we are interested in the existence of a $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that $$(u(T),\varphi(t))_H -\int_0^T \langle \varphi'(t), u(t) \rangle_{V^*,V} + \int_0^T ...
3
votes
1answer
112 views

Checking initial data in parabolic PDE with no control on time derivative

It is possible to define a weak solution of a parabolic PDE $$u_t - Au = f$$ $$u(0) = u_0$$ as $u \in L^2(0,T;H^1)$ such that $$-\int_0^T\int_\Omega u(t)\varphi'(t) + \int_0^T\int_\Omega ...
0
votes
0answers
63 views

Closure of partial differential operators on $L^2(\Omega)$

Let $\Omega\subset\mathbb{R}^2$ open set. Consider an $\textit{uniformly elliptic}$ second order differential operator on $L^2(\Omega,\mathbb{C})$ $$ H=\sum_{|\alpha|\le 2}C_\alpha\partial^\alpha $$ ...
3
votes
0answers
144 views

Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$

Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required. Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b < ...
1
vote
0answers
110 views

A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form $$u_t(t) - ...
1
vote
1answer
209 views

Are smooth functions dense in the space $\{u \in H^1(Q) \text{ with } \Delta_\Gamma u \in L^2(Q)\}$?

Define $$Q = \bigcup_{t \in (0,T)}\Gamma \times \{t\}$$ where $\Gamma$ is a compact (without boundary) hypersurface. Assume whatever smoothness is required. Define $L^2(Q) := L^2(0,T;L^2(\Gamma))$ ...
4
votes
1answer
204 views

Bounding $\lVert{u}\rVert_{C^0([0,T];V)} \leq C\left(\lVert{u}\rVert_{L^2(0,T;V)} + \lVert{u'}\rVert_{L^2(0,T;H)}\right)$?

Define $W(X,Y) = \{ u \in L^2(0,T;X) \mid u' \in L^2(0,T;Y)\}$ where $u'$ is the usual weak derivative. Let $V \subset H \subset V^*$ be a Hilbert triple. If $u \in W(V,V)$ (so $$u \in L^2(0,T;V) ...
-3
votes
1answer
157 views

$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?

Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at ...
0
votes
0answers
39 views

Well-posedness of a Stefan problem with Faedo-Galerkin approach

Given a domain $\Omega$ which is divided by $\Omega_1(t)$ and $\Omega_2(t)$ and the interface $\Gamma(t)$, does anyone have a reference to where a Stefan problem of the type $$\frac{d}{dt}H(u) - ...
2
votes
0answers
94 views

Almost a Green formula

Let $\Omega$ be the half-space $\mathbb{R}^{n-1}\timesĀ \{ x_n>0 \}$, let $v \in L^2(\Omega)$ and $\phi\in \mathcal{C}^{\infty}(\overline{\Omega})$ with compact support in $\overline{\Omega}$. What ...
1
vote
1answer
86 views

well-posedness of heat equation with Neumann BC and periodic data

On a domain $\Omega$ with $f \in L^2(0,T;H^{-1})$ such that $f(0) = f(T)$, consider $$u_t - \Delta u = f\quad\text{on $\Omega$}$$ $$\frac{\partial u}{\partial \nu} = 0\quad\text{on $\partial\Omega$}$$ ...
1
vote
0answers
101 views

Weak periodic solution of parabolic PDE

Take $$ u_t(t) + A(t)u(t) = f(t), $$ $$ u(0) = u(T), $$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
-1
votes
1answer
175 views

Why can't I get global existence to linear PDE in this way? [closed]

For any $n > 0$, standard theory implies there is a unique $u_n \in L^2(0,n;V)$ with $u_n' \in L^2(0,n;V^*)$ such that $$u_n' + Au_n = f\quad\text{as an equality in $L^2(0,n;V^*)$}$$ $$u_n(0) = ...
2
votes
1answer
182 views

Inequality in the Sobolev space $H^1$

I've found the following inequality $$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ...
0
votes
1answer
138 views

A Poincaré-type inequality with logarithmic function

For any function $f(x)$ we denote $\bar{f}:=\frac{1}{\Omega}\int_\Omega f(x)\,dx$. Let $\Omega\subset \mathbb{R}^n$ be a bounded smooth domain and $u(x)> 0$ be a smooth function defined on ...
1
vote
1answer
201 views

Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?

Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous? I define $H^{\frac 1 ...
1
vote
2answers
84 views

References for well-posedness of weak solutions to Stefan problem

Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)? ...
1
vote
2answers
162 views

Interior Schauder estimates with weights

Suppose we have $u(x)\in H_2^{loc}(\Omega_{\rho})$, where $\Omega_{\rho}=\{x\in \mathbb{R}^n, |x|>\rho\}$, and in $\Omega_{\rho}$, $u$ satisfies the equation $$ \Delta u-V(x)u=0, $$ where $V$ is a ...
2
votes
1answer
143 views

If $f \in H^{\frac 12}$ and $\varphi$ is Lipschitz, is $f\varphi \in H^{\frac 12}$ (on a Lipschitz hypersurface)?

Let $M$ be a bounded hypersurface. Let $f \in H^{\frac 12}(M)$ and let $\varphi\colon M \to \mathbb{R}$ be a Lipschitz function. When $M=\Omega \subset \mathbb{R}^n$ an open domain, we know that the ...
2
votes
2answers
152 views

property of local sobolev space

The local Sobolev space,defined as $W^{k,p}_{loc}(\Omega)$, is the space such that for any $u \in W^{k,p}_{loc}(\Omega)$ and any compact $V\subset \Omega$, $u \in W^{k,p}(V)$. I am just wondering if ...
3
votes
2answers
175 views

Sobolev spaces on boundaries

Consider the Sobolev space $W^{s,2}=H^s$ for $s=\frac{1}{2}.$ Let $\Omega \subset \mathbb{R}^n$ be an open set with boundary $\partial\Omega$. I have seen two definitions of the space ...
3
votes
0answers
193 views

Alternative representations of Sobolev space

Is there a way to represent a Sobolev space as the image of a fractional integral operator over an $L^p$ Lebesgue space? Yes, as it was comment, there is an answer for that in the book "Singular ...
3
votes
1answer
260 views

traces of sobolev spaces under additional assumptions

Let $p\in [1,\infty]$, $\Omega$ an open bounded domain with (smooth, if necessary) boundary $\partial \Omega$. Is there a subspace $X\subset L^p(\Omega)$ - a simply describable space, ideally a ...
1
vote
1answer
168 views

Examples of functions in $W^{k,p}(\Omega)$ with exact smoothness

Please give, explicitly, a function $f:\Omega\mapsto\mathbb{R}$ such that $f\in W^{k,p}(\Omega)$ but $f\notin W^{s,p}(\Omega)$ for $s>k$. Here $\Omega$ can be a subset of $\mathbb{R}^n$ with ...