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65 views

Estimate divergence by gradient in H1

I am currently trying to fully understand the stationary Stokes equations of incompressible fluid. In the mixed form (homogeneous boundary data), for $\Omega\subset\mathbb{R}^d$, $d\in\{2,3\}$, a ...
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1answer
55 views

How to establish the estimate?

I consider the inequality as follows: let $a>0,b>0$, and satisfy $$2a^2-b^2\leq C(1+a).\tag{1}$$ If we assume that $b\leq a$, then there exist a constant $C_1>0$ such that $$a\leq ...
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128 views

Interpret the terms in Strang's second lemma

The second lemma of Strang states that for a certain choice of $V_h$, $a$, $u$ and $f$ there exists a $c>0$ such that $$||u-u_h|| \leq c (\inf_{v\in V_h} ||u-v|| + \sup_{v\in V_h} ...
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86 views

Inverse estimate of gradient of Sobolev function

I need an estimate for $\| \nabla w\|_{L^2{(\Omega \subset \mathbb{R}^n)}}$, such that it is $< c\| w\|,\ w \in H_0^1(\Omega)\ $. Is this possible?
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181 views

Lemma 3.3 in the book of the Fanghua Lin

Lemma 3.3 Suppose $u \in H^{1}(\Omega)$ satisfies $$\int_{B_r(x_0)}|Du|^2 \le M r^{^\mu} \quad \mbox{for any} \ \ B_r(x_0) \subset \Omega,$$ for some $\mu \in [0,n)$. Then, for any $\Omega' ...