# Tagged Questions

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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### $L^\infty$ bound on solutions of linear parabolic equations

We work on a closed Riemannian manifold $M$. Let $u$ and $v$ be the non-negative weak solutions of $$au_t - 2d\,\Delta au = cv - f$$ $$bv_t - d\,\Delta bv = f$$ $$u(0)=u_0, \quad v(0)=v_0$$ where $f$ ...
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### Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$, $\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$ where the scalar complex function ...
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### Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure. Assume $\partial\Omega=\partial\Omega'$. Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
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### Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow \begin{cases} ‎\Delta^2u=‎‎\lambda ‎‎\dfrac{u}{|x|^4}‎‎+u^{‎p}‎ & \mathrm{in}‎\hspace{...
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### Fractional Sobolev spaces and extension by zero

The all-wise Wikipedia suggest (https://en.wikipedia.org/wiki/Sobolev_space#Extension_by_zero) that for $s > 1/2$ any function from $H^s_0(\Omega)$ can be extended by zero to whole $\mathbb{R}^n$ (...
I am escalating this question from SE, as I have been unable to obtain any guidence in relation to a possible solution. Some context, I am working with weighted Sobolev spaces of the form $W^{m,2}(I,... 1answer 187 views ### Completion of$C_0^{\infty}(\mathbb{R}^N)$with norm$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $I have a question that I could not find it any where. Is the completion of$C_0^{\infty}(\mathbb{R}^N)$with the respect to norm $$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{... 0answers 227 views ### The Spectrum of certain differential operators We fix a Hilbert space isomorphism \phi:H^{1}\to H^{2}. Here by H^{s},\;s=1,2,\; we mean the sobolev space on \mathbb{R}^{2} or S^{2}. We consider the following polynomial vector field on ... 1answer 124 views ### Extension by zero in Sobolev spaces Let \Omega be an open bounded set of R^n, and let \omega be an open subset of \Omega s.t \overline{\omega} \subset \Omega. For f\in H_0^1(\omega), it is known that the extension of f to ... 0answers 42 views ### The Best Korn's constant for bounded deformation I am studying the following version of Korn's inequality. For u\in BD(\Omega), BD denotes the bounded deformation space, we have, there exists a r(u)\in \operatorname{ker}\mathcal E such that$$... 1answer 78 views ### Does this time-dependent trace space have a name? This question is a follow up to this question. Let$\Omega \subset \mathbb{R}^d$be an open connected set. For each$t\in \mathbb{R}^+$let$u_d:\partial\Omega \to \mathbb{R}$be in$H^{1/2}(\...
The following general definition of subharmonic function comes from the classical text book [elliptic partial differential equations of second order] by Gilbarg and Trudinger. We call a function $u$ ...