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0
votes
1answer
122 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
170 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
94 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 ...
3
votes
0answers
91 views

Existence and smoothness of convolutions of distributions in Sobolev spaces

Let $f\in H^{s_1}(\mathbb{R}^n)$ and $g\in H^{s_2}(\mathbb{R}^n)$, where $s_1, s_2 \in \mathbb{R}$ and can be positive or negative. It is easy to show that $f *g$ is defined pointwise when ...
5
votes
1answer
248 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) = ...
3
votes
1answer
214 views

$H^s(\mathbb T)$ is a Banach algebra for $s>1/2$

I have not managed to find a reference for the following fact: $H^s(\mathbb T)$ is a Banach algebra for $s>1/2$. In particular, I need reference for the following inequality: $$ \|uv\|_{H^s} ...
2
votes
1answer
79 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
120 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 ...
4
votes
0answers
258 views

Which functions can be approximated by piecewise constant functions?

Let $\Omega \subset \mathbb{R}^d$ be a connected and bounded domain. We call a function $f\in L^\infty(\Omega)$ nice if for each $\epsilon>0$ there exist $n\in \mathbb{N}, a_1,\dots,a_n \in ...
4
votes
2answers
240 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
108 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
2answers
291 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
77 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
406 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 ...
0
votes
1answer
198 views

reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$

I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...
1
vote
0answers
64 views

Is $\partial^\alpha$ a map $H^{s,p}(\mathbb R^N,F)\to H^{s-|\alpha|,p}(\mathbb R^N,F)$?

More precisely, the question is formulated as follows. Let $F$ be an arbitrary Banach space, especially not having the UMD−property. Let $N\in\mathbb N$ and $s\in\mathbb R$ and $1\le p\le +\infty$ . ...
30
votes
1answer
726 views

For which maps $S^1\to S^1$ is the winding number defined?

There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number: • Continuous maps: Using the unique path lifting property of the universal covering map $\mathbb R\to ...
1
vote
1answer
122 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)) = ...
10
votes
1answer
187 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
57 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
124 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)\\ ...
1
vote
2answers
119 views

reference needed for sobolev type estimates

I'm reading a paper and the authors applied the following sobolev type estimates $$ ||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2} $$ for $\alpha>\frac{1}{4}$, where $v$ ...
6
votes
2answers
333 views

Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...
5
votes
1answer
248 views

Embeddings of Sobolev spaces

Let $s_1,s_2\in \mathbb R$ such that $-\frac12<s_1\le s_2$. There exists $C>0$ such that for all smooth functions $w$ , for all $r>0$, $$\operatorname{supp} w \subset(-r,r)\Longrightarrow ...
2
votes
0answers
74 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
120 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 ...
3
votes
0answers
171 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
122 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) - ...
3
votes
1answer
186 views

Is this integration by parts legitimate?

Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I_2[g]$ $$ f(x)=\int_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy, $$ which solves $-\Delta f=g$ (up to positive normalizing constants, which I ...
2
votes
1answer
260 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 $L^2(Q) := L^2(0,T;L^2(\Gamma))$ ...
3
votes
0answers
162 views

Trace Theorem for $p=\infty$

I am considering the Sobolev space $W^{1,\infty}(\Omega)$ on a bounded Lipschitz domain $\Omega \subseteq \mathbb{R}^2$. I am wondering whether the trace theorem holds in this case with constant one ...
4
votes
1answer
227 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
173 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 ...
2
votes
0answers
98 views

Almost a Green formula

Let $\Omega$ be the half-space $\mathbb{R}^{n-1}\times \{ x_n>0 \}$, let $v \in L^2(\Omega)$ and $\phi\in \mathcal{C}^{\infty}(\overline{\Omega})$ with compact support in $\overline{\Omega}$. What ...
1
vote
1answer
95 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 $$u_t - \Delta u = f\quad\text{on $\Omega$}$$ $$\frac{\partial u}{\partial \nu} = 0\quad\text{on $\partial\Omega$}$$ ...
3
votes
1answer
234 views

Equivalence of negative Sobolev norm of derivative to $L^2$-norm

Let $S:=(0,1)^2$ be the unit square in $\mathbb{R}^2$, and let $M:=\{u\in L^2(S)\mid \int_S u=0\}$ be the space of (real-valued) $L^2$-functions with mean value zero. On $M$ we can consider the ...
1
vote
0answers
117 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
196 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) = ...
3
votes
1answer
206 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
152 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 ...
3
votes
2answers
299 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
98 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)? ...
2
votes
2answers
198 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
158 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
223 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
232 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 ...
1
vote
1answer
84 views

Which rate of growth of the Sobolev norms guarantees analyticity?

Let $u\in C^\infty(\mathbb T^k)$, where $\mathbb T^k$ is the $k$-dimensional torus. (Equivalently, $u\in\mathbb R^k$ and $u$ is $2\pi$-periodic with respect of each argument.) We define the semi-norm ...
3
votes
1answer
136 views

sub and super-levelset regularity for Sobolev functions

I'm wondering if there are known results about the "regularity" (in some sense to be determined) of sub and super levelsets of Sobolev functions $u\in W^{1,p}(\mathbb{R}^d)$. More precisely: Assume ...
4
votes
0answers
208 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 ...
4
votes
1answer
97 views

Examples of optimal ultracontractivity estimates for a Markovian semigroup $T_t$ that do not depend polynomialy on $t$

Let $(X,\mu)$ be a measure space and $T_t : L_2(\mu) \to L_2(\mu)$ for $t \geq 0$ a symmetric Markovian semigroup. Local ultracontractivity estimates of the form: $$ \| T_t : L_p(\mu) \to L_q(\mu)\| ...