Questions tagged [sobolev-spaces]
A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.
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Equivalent norms of fractional Sobolev spaces on bounded Lipschitz domain
Let $s>0$, $1<p<\infty$ and let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. Set $H^{s,p}(\Omega)=\{u\in L^p(\Omega):\exists\tilde u\in L^p(\mathbb R^n),\tilde u|_{\Omega}=u,(I-\...
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Weighted version of the Gagliardo Nirenberg inequality
I'm searching for a weighted Gagliardo-Nirenberg inequality similar as in https://arxiv.org/pdf/1307.1363.pdf where the weight is a power of the last component. Is there an inequality of the form
$$
\...
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$f\in C(B_1)\cap W^{1,2}(B_1\setminus \{f=0\})$ implies $f\in W^{1,2}(B_1)$?
In a paper I am writing I need to show that a certain real-valued function $f\in C^0(B_1)$ belongs to the Sobolev space $W^{1,2}(B_1)$ ($B_1$ is the unit ball). So far I have been able to show that ...
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Is this a well known space? Perhaps homogeneous Sobolev-like space?
The homogeneous Sobolev space $\dot H^s(\mathbb{R}^n) $ is often defined as the closure of $\mathcal{S}(\mathbb{R}^n)$ under the norm
$$ || |\omega|^s \widehat{f} ||_{L^2(\mathbb{R}^d)} =\int_{\...
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Noncompactness of the Sobolev embedding in the critical exponent case
Let $\Omega \subset \mathbb R^n$ be a bounded domain with a Lipschitz boundary and $n > p \ge 1$.
It is well known that up to the critical exponent $p^* = pn/(n − p)$, i.e. $q < p^*$, the ...
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Localized estimate for divergence free vector field
Suppose $\Omega \subset \mathbb{R}^3$ is a simply connected Lipchitz domain. For a divergence free field $w\in [L^2(\Omega)]^d$, it is well known that there exists a vector field $v\in [W^{1,2}(\Omega)...
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Regularity of the Jacobian of a $W^{2,n}$ Sobolev mapping
Given a mapping in the Sobolev space $f\in W^{2,n}_{\rm loc}(\mathbb{R}^n,\mathbb{R}^n)$ I would like to know what is the
Sobolev regularity of the Jacobian $J_f=\operatorname{det} Df$.
It is well ...
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Dependence of Sobolev embedding theorem constant on smoothness
Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
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Equivalent characterization of weak derivative in Bochner space
Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff
$$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...
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Friedrich's second inequality for functions with zero average
Friedrich's second inequality (or Maxwell Estimates or Gaffney’s inequality in the literature) is referred as follows: for all $\mathbf{u} \in H^1(\Omega)^2$ satisfying either $\mathbf{n} \cdot \...
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finite dimensionality of a subspace of a Banach space
Let $H$ be the space of measurable functions on $(0,1)$ such that
$$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$
Let $C>0$ be a constant. Suppose that $W \...
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Sobolev embedding theorems on manifolds
I had asked the following question on math.stackexchange but did not get any response:
I'm looking for a reference which states the Sobolev embedding theorems on Riemann manifolds for fractional ...
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Sobolev spaces and geometry
This is a very naive question, is there a way to geometrically understand Sobolev spaces without going through analysis and PDE's? To my knowledge, Sobolev spaces where created precisely to study PDE'...
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Blow up for certain classes of distributions
Let $\mathbb D$ be the open unit disc centered at the origin and let $u \in H^{-N}(\mathbb D)$ be a distribution for some natural number $N>0$. Suppose that $$u|_{\mathbb D\setminus \{0\}} \in C^{\...
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A compact embedding claim
Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms
$$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$
Let $H_2$ be the weighted Sobolev space with the ...
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Is the local maximal function bounded from $W^{1, 1}$ to $L^1$?
Let $f \in W^{1, 1} (\mathbb R^d)$. For every $\varepsilon > 0$, we consider the local maximal function $M_\varepsilon f: \mathbb R^d \to \mathbb R$, defined by
$$M f_{\varepsilon} (x) = \sup_{r \...
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Norm inequality for the inclusion $L^2(\partial \Omega)\hookrightarrow H^{-1/2}(\partial \Omega)$
Let $\Omega \subset \mathbb{R}^3$ be a lipschitz domain. We then have the trace operator $\tau : H^1(\Omega) \to L^2(\partial \Omega)$ and can define the space $H^{1/2}(\partial \Omega) := \tau(H^1(\...
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Approximation of monotone Sobolev functions
Let $f\in W_{loc}^{1,2}(\mathbb R^2)$ be a continuous monotone (real valued) function (monotone in the sense that the maximum and minimum of $f$ in a precompact open set are attained at the boundary)....
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On norm of the Sobolev space $H^2(\Omega)$, $\Omega \subset \mathbb{R}^n; n \geq 2$
Let the Sobolev space $H^2(\Omega)$ be defined with the norm $\|u\|_{H^2(\Omega)}=\Big(\sum_{|\alpha|\leq 2})\|D^{\alpha}u\|^2_{L^2(\Omega)}\Big)^\frac{1}{2}$.
I have found in several research ...
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Dual norm for weighted Sobolev space
Consider a "weight" function $\rho: \mathbb{R}^{d} \rightarrow \mathbb{R}$ and the following weighted Sobolev norm:
\begin{equation}
\|h\|_{\dot{H}^{1}(\rho;\mathbb{R}^{d})}:=(\int_{\mathbb{...
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About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"
I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1.
We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
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Sobolev-type estimate for irrational winding on a torus
Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
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Self-adjointness of fractional laplacian
Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
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Non-uniqueness of solutions to a simple nonlinear elliptic PDE in $\mathbb R^n$
My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice ...
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Blowup of Sobolev norms in approximating a non-absolutely continuous function
Let $f: [0, 1] \to \mathbb R$ be a continuous function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly. Is it true that if $f$ fails to be absolutely ...
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Existence of a curve of finite length on the image of an embedding which is Sobolev
Suppose that we have an embedding $f:\mathbb{R}^2\to\mathbb{R}^3$ which belongs in the Sobolev space $W^{1,p}_{loc}(\mathbb{R^2},\mathbb{R}^3)$ for some $p>2$. Is it true then that for any two ...
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Why do we assume that $\mathcal{A}$ is an algebra in this 2003 paper of Bobkov and Tetali?
In Bobkov and Tetali - Modified Log-Sobolev Inequalities, Mixing and Hypercontractivity (extended version Modified Logarithmic Sobolev Inequalities in Discrete Settings), at the beginning of section 3,...
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Sobolev inequalities in weighted Sobolev spaces
My definition for "weighted Sobolev space" $W^{1,p} _w (\Omega)$ is just a set of functions with the property that they admit distributional derivatives and that
$$ \int_\Omega |f|^p (x) w(x)...
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A kind of weak convergence for Sobolev spaces with zero on boundary
Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$. Is it true that if $(\varphi_n)_{n\geq 1}\subset C^{\infty}_c(\Omega)$ with $\varphi_n\to \...
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Extending Hölder functions
I originally asked this question on MathStackExchange some time ago, but it seems that MathOverflow would be more appropriate. Essentially, I would like to find references for extension theorems for (...
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Weak convergence in $H^{1}$ implies different convergence in $L^{p}$?
Suppose I have a sequence $\{f_{n}\}_{n\in \mathbb{N}} \subset H^{1}(\mathbb{R}^{d})$ which converges weakly to $f$ in $H^{1}(\mathbb{R}^{d})$, in the sense that $\langle f_{n},\varphi \rangle_{L^{2}}+...
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On a Poincaré inequality with weight
Let $\Omega$ be a bounded convex (non-empty) open subset of $\mathbb{R}^n$ ($\Omega$ can be as smooth as you like). Let also $p, q > 1$ be conjugate exponents.
Is it true that there exists a ...
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When is $W^{1,p}(\Omega)$ a Banach algebra?
Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$.
My question: knowing that $f,g\in W^{1,p}(\Omega)$ for what $p$ can we conclude that $f\...
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Estimate for the gradient of solutions in an elliptic differential equation in a Sobolev space
Let $\Omega$ be a bounded or unbounded domain in $\mathbf R^{3}$ with a smooth boundary $S$ and a normal vector given by $n$. Now, we consider the following second-order elliptic problem with Neumann ...
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A question about equivalence of weighted Sobolev space norm in S. Benzoni-Gavage and D. Serre's book
This question may not be at the research level, but it has really bothered me for a long time. The following space is used for handling initial boundary value problem for first order hyperbolic ...
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Express Dirichlet energy $E_\mu(f) := \int \|\nabla f(x)\|^2 d\mu(x)$ in terms of Fourier information alone
Let $\mathbb R^d$ and let $\mu = p(dx)$ be a probability distribution thereupon, with density $p$ (which maybe assumed bounded, etc.). For a continuously differentiable function $f:\mathbb R^d \to \...
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Uniqueness in interpolation of Hilbert spaces
I am wondering under what condition it is true that for Hilbert spaces $H$ and $H_0$ such that $H \hookrightarrow H_0$ there is uniqueness in the existence of a Hilbert space $H_1 \hookrightarrow H$ ...
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Books about capacity theory
While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for ...
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Is there any bilinear Poincaré/Sobolev inequality?
Is the following, I call it bilinear Poincaré inequality, true?
Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in ...
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On the compact embedding of Sobolev space
In dimension three, we know that the Sobolev space $W^{\frac{13}{11},11}(D)$ is compactly embedded into $W^{1,11}(D)$, where $D$ is a bounded domain in $R^3$ with smooth boundary. My question is: Does ...
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Are weak solutions and mild solutions for linear parabolic equations equivalent in $L^{q}([0,T],L^p(\Omega))$ with $1<q<\infty$, $1<p \leq 6/5$?
I have looked through some MO and ME posts, and the common opinion is that weak and mild solutions are equivalent for "many" cases of linear parabolic equation.
However, detailed proofs can ...
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Hölderness of the inverse to a $W^{1,p}$-homeomorphism (with additional conditions) of a Lipschitz domain
Let $\Omega_1 \subset \mathbb R^2$ be a bounded simply-connected Lipschitz domain, and $f: \bar \Omega_1 \rightarrow \bar \Omega_2$ be a homeomorphism, which is a diffeomorphism on $\Omega_1$ such ...
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Understanding the Bochner space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ in terms of the Fréchet derivative
In the context of linear parabolic equations, the Sobolev space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ appears all the time. Here, $U$ is some bounded region of $\mathbb{R}^n$ and $1<p,q<\infty$.
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Relationship between $p$-capacity and Riesz $s$-capacity of a set
What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set?
Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
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Introduction to free boundary problems (that are not Stefan problems)
Could someone recommend some notes/papers that deal with existence/regularity of free boundary problems arising from parabolic equations (excluding Stefan type equations)?
I am thinking of eg. ...
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Riesz’s representation theorem in a weak form
Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^N$ $(N\geq 3)$, $\phi\in H_0^1(\Omega)$ is a solution of $$
\begin{cases}\Delta \phi+ \phi=h & \text { in } \Omega, \\ \phi=0 &...
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Extension Operator for $W^{1,\infty}(U,X)$
I am reading through some lectures on Sobolev spaces and the vector-valued (or Banach space valued) version of them. At this moment I am very interested in extension operators for the vector-valued ...
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Validity of Hölder inequality for the homogeneous Besov spaces $\dot{B}^0_{1,2}(\mathbb{R}^n)$ and $\dot{B}^0_{2,2}(\mathbb{R}^n)=L^2(\mathbb{R}^n)$
I am looking at Corollary 1. in p.244-245 of the book
"Sobolev Spaces of Fractional Order,
Nemytskij Operators,
and Nonlinear
Partial Differential Equations" (1996) by Thomas Runst
Winfried ...
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Caffarelli-Kohn-Nirenberg-type inequality with nonradial weight
The Caffarelli-Kohn-Nirenberg inequalities are a set of inequalities generalizing the Gagliardo-Nirenberg inequalities and are of the form
$$\||x|^\gamma u\|_{L^p} \leq C\||x|^\alpha \nabla u\|_{L^q}^...
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Controlling the tensor product of functions in $H^1$ with lower derivatives
Given some function $\phi:D\to\mathbb{R}$, with $D\subseteq \mathbb{R}^d$. I was wondering whether we can find an estimate of the form
$$ \|\phi\otimes\phi\|_{\dot{H}^1(D\times D)} \lesssim \|\phi\|_{\...
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