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        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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        Existence and regularity for fractional Poisson-type equation

        According to Theorem 2.7 in the paper https://arxiv.org/pdf/1704.07560.pdf, we have the following classical results. Let $s \in (0,1)$ and $1<p<\infty$. Then for any $F \in L^p(\mathbb{R}^n)$ ...
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        Specific Bounds on Divergence operator in Sobolev setting

        Given that $\vec{u}=(u_1,u_2)\in H^1(\Omega)\times H^1(\Omega)$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded with Lipschitz boundary, does there exist specific inequalities which links $\...
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        Regularity in Orlicz spaces for the Poisson equation

        I see the following Lemma in :Regularity in Orlicz spaces for the Poisson equation | SpringerLink:(2007) $$\Delta u=f \quad \quad \quad \quad (1)$$ Lemma 2: There is a constant $N_1 >1$ so that ...
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        The Poincaré inequality for the Sobolev space on a domain with a non Lipschitz boundary

        Let $\Omega$ be a bounded open Lipschitz domain in $\mathbb{R}^{d}, d \geq 3$. Assume that $L$ is a straight segment such that $L \subset int(\Omega)$. Let $v \in V:= \overline{ \{ \phi \in C^{\infty}...
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        Interpolation inequalities involving mean curvature operator

        Are there any interpolation inequalities (for example, of Gagliardo-Nirenberg type) involving the mean curvature operator $$\mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)$$ (in any ...
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        Unboundedness of the Sobolev norm of a sequence : Does it follow from Sobolev embedding?

        Let $\{f_n\}$ be a sequence of functions that are continuous and lying in $H^k(\mathbb{R}^m)$. Assuming $k>\frac{m}{2}$, and if $f_n \to f$ pointwise, where $f\in H^k(\mathbb{R}^m)$, such that $f$ ...
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        A specific problem on : Can bounding the Sobolev norm, bound a higher derivative?

        Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|_{...
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        Density on a specific functional space.

        I have a question about density. It's probably trivial but I am just learning functional analysis so nothing is trivial to me. Here is my question. Let $$ \mathcal{X}\colon=\mathcal{H}^1(0,1;\mathbb{...
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        Making the Fourier transform quantitative

        I am undergraduate Physics student and understand that this is a professional mathematics forum. But due to perhaps broader interest, I hope this question is suitable for this website. I understand ...
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        Fractional embedding inequality with $L^{\infty}$ norm

        Here we consider the fractional Sobolev spaces and suppose $u$ is a vector function in $\mathbb R^2$. For $q>2$, is the following always true? $$\Vert Du \Vert_{L^{\infty}(\mathbb R^2)} \leq C\Vert ...
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        Bounded deformation vs bounded variation

        Let $BV(\mathbb R^n; \mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(\mathbb R^n;\mathbb R^n)$ the space of functions with bounded deformation. They are made ...
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        Removable set for Sobolev space

        It is well known that if $\Omega\subset\mathbb{R}^{N}$ open, $F\subset\Omega$ closed, such that $\mathcal{H}^{N?1}(F)=0$,where $\mathcal{H}^{N?1}$ denotes (N-1) dimensional Hausdorff measure, then $W^{...
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        $f_n$ is bounded in $C(0,T;H^2(0,L))$ so is $f_n^p$?

        Let $1<p<\infty$, and $f_n$ be a bounded sequence in $C(0,T;H^2(0,L))$. It looks obvious to me that $f_n^p$ is also bounded in $C(0,T;H^2(0,L))$. When we take the derivative of $f^p(t)$ twice we ...
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        Positive splitting of Sobolev convergence

        Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions wuch that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ ...
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        Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$?

        Let $X$ be a Banach space, $T(t)$ be a strongly continuous semigroup on $X$, and $f\in L^1(0,\tau;X)$. It has been implied that the integral $$v(t)=\int_0^t T(t-s)f(s)ds,\quad t\in [0,\tau]$$ is not ...

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