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        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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        139 views

        Question on Sobolev spaces in domains with boundary

        Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Define the Sobolev norm on $C^\infty(\bar \Omega)$ $$||u||_{W^{1,2}}:=\sqrt{\int_\Omega (|\nabla u|^2+u^2)dx}.$$ ...
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        Gagliardo-Nirenberg inequality for periodic functions?

        I am interested in Gagliardo-Nirenberg type inequality (see https://en.m.wikipedia.org/wiki/Gagliardo%E2%80%93Nirenberg_interpolation_inequality) for functions in the space $$H^1_T(\mathbb{R}^n)=\...
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        $W^{k,p}$ and Holder regularity for linear elliptic systems with Neumann boundary data

        I'm looking for a text or paper that discusses regularity in the Sobolev and Holder sense for general linear elliptic systems of PDEs on bounded domains with Neumann boundary data. The book by ...
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        Using weak maximum principle to prove continuous dependence of the boundary data?

        I am currently looking at the following ingomogenous Dirichlet problem over an open, bounded domain $\Omega \subset \mathbb{R}^2$ with continuous boundary: \begin{align} \begin{cases} -\operatorname{...
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        1answer
        103 views

        Embedding of weighted sobolev space with exponential weights

        In the book by Bensoussan and Lions, they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. ...
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        Question: can we claim that $W^{1,p}(\Omega) \subset L^1(\Omega) \subset W^{-1,p}(\Omega)$?

        Let $\Omega$ be a compact manifold in $\mathbb R^2$. For $1 \leq p \lt 4/3$ can we claim that $W^{1,p}(\Omega) \subset L^1(\Omega) \subset W^{-1,p}(\Omega)$ with the first inclusion being ...
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        Is the normalized derivative of a holomorphic function Sobolev?

        This question is a cross-post from MSE. it is also a special case of this question. Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$, and let $f:B \to \mathbb{C}$ be holomorphic on the interior $B^o$, and ...
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        1answer
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        Global estimate to an L1 function whose Laplacian is a bounded measure

        Pretty simple question: Suppose that $u \in L^1(\mathbb{R}^N)$ is such that $\Delta u \in \mathcal{M}(\mathbb{R}^N)$ (i.e., $\Delta u$ is a bounded Radon measure). Does $\nabla u \in L^1(\mathbb{R}^N)$...
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        Harmonic functions vanishing on the boundary and distance function asymptotics

        Let $\Omega \subset \mathbb R^N$ be a $C^2$ domain. Let $u$ be a function such that $u \in W^{2,2}(\Omega)$ and $u = \Delta u = 0$ on $\partial \Omega$. Is it true that $$ c \le \frac{u}{[\mathrm{dist}...
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        Can we approximate harmonic maps which are a.e. orientation-preserving with maps which preserve orientation globally?

        Let $\mathbb{D}^n$ be the closed unit disk, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be harmonic; suppose that $n \ge 2$, and that $\det df >0$ a.e. on $\mathbb{D}^n$. Are there harmonic maps $\...
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        77 views

        Discrete Sobolev embedding

        It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$ Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...
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        198 views

        Regularity result for the boundary value problem for the heat equation

        Let $\Omega$ be an open bounded subset of $\mathbb R^N$. Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$ Consider the following boundary value problem for the heat equation: ...
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        150 views

        Is polar decomposition of a smooth map Sobolev?

        Motivation: Let $\mathbb{D}^2$ be the closed unit disk. I am studying the "elastic energy" functional $E(f)=\int_{\mathbb{D}^2} \text{dist}^2(df,\text{SO}_2)$, where $f \in C^{\infty}(\mathbb{D}^2,\...
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        129 views

        The distributional gradient of the closest isometry to the differential of a smooth map

        The setting-a "linear algebra" fact: Let $A$ be a real $n \times n$ matrix, and suppose that $\det A<0$ and that the singular values of $A$ are distinct. Then, there exist a unique matrix $Q(A) \...
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        84 views

        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 ...

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