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293 views

Normal form transformation

Hi, my question is related to normal form transformations...One of the papers I would like to understand is Wu, S., "Well-posedness in Sobolev spaces of the full water wave problem in 2-D" where the ...
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2answers
365 views

Sobolev space: probably simple ode…

I am trying to solve for $y(x)$ in terms of $f(x)$ in a convenient space (eg. $\dot{H}^2(\mathbb{T})$-zero mean). Here is the ode: $y(x)+y(x)y'(x)=f(x)$. I think a contraction mapping argument will ...
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1answer
181 views

Do extracted weak $H^{1,2}$-limits and $C^0$-limits coincide?

Let $I$ be a bounded interval and consider a sequence $(u_k)$ in $H^{1,2}(I)$ (usual Sobolev space). Suppose furthermore, that the sequence $(u_k)$ is bounded in $H^{1,2}(I)$. Then, by Rellich, we can ...
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2answers
1k views

Sobolev imbedding on Riemannian manifolds

Let $(M, g)$ be a non-compact smooth Riemannian manifold of dimension $n \ge 2$, and $G$ a subgroup of the isometry group of $(M,g)$, say with $G$ contained in the component of the identy. Let ...
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1answer
388 views

Sobolev imbedding

It is known that, for $n \ge 3, 2 < p< 2^*$, the imbedding $H^1(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n)$ is not compact. Let $G=O(n_1) \times O(n_2)\times\cdots\times O(n_k)$, with ...
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4answers
987 views

Trace theorem for $C^{k,1}$ domains

What are the best results on (Sobolev space) trace theorems for $C^{k,1}$ domains? For $k=0$, e.g., when the domain is Lipschitz, from e.g. the works of Martin Costabel and Zhonghai Ding, it is known ...
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1answer
700 views

Sobolev-Slobodeckij spaces for p=infinity

For $1\leq p<\infty$ an approach to define fractional Sobolev spaces is by Sobolev-Slobodeckij spaces a generalisation of Hölder continuity. For example letting $U\subset\mathbb{R}^n$ then, $ ...