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Posts Tagged ‘sobolev’

Characterizations of Sobolev Spaces

October 26, 2012 1 comment

I will present here a few useful characterizations of Sobolev spaces. The source is Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis. Consider {u \in L^p(\Omega)} with {1 < p \leq \infty}. We denote by {p'} the conjugate of {p}, i.e. {\displaystyle \frac{1}{p}+\frac{1}{p'}=1}. Then the following properties are equivalent:

  • (i) {u \in W^{1,p}(\Omega)};
  • (ii) there exists a constant {C} such that

    \displaystyle \left| \int_\Omega u \frac{\partial \varphi}{\partial x_i} \right| \leq C \| \varphi\|_{p'} ,\ \forall \varphi \in C_c^\infty(\Omega),\ \forall i=1..N;;

  • (iii) there exists a constant {C} such that for all {\omega \subset \subset \Omega} and all {h \in \Bbb{R}^N} with {|h|<\text{dist}(\omega,\partial \Omega)} we have

    \displaystyle \| \tau_hu-u\|_{L^p(\omega)}\leq C|h|.

    (note that {\tau_hu(x)=u(x+h)} makes sense for {x \in \omega} and {|h|<\text{dist}(\omega,\partial \Omega)}. Furthermore, we can take {C=\|\nabla u\|_{L^p(\Omega)}} in (ii) and (iii). If {\Omega=\Bbb{R}^N} we have

    \displaystyle \|\tau_hu-u\|_{L^p(\Bbb{R}^N)} \leq |h|\|\nabla u\|_{L^p(\Bbb{R}^N)}.

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One dimensional trace of a Sobolev function

October 7, 2012 Leave a comment

Check that the mapping {u\mapsto u(0)} from {H^1(\Bbb{R})} to {\Bbb{R}} is a continuous linear functional on {H^1(0,1)}. Deduce that there exists a unique {v_0 \in H^1(0,1)} such that

\displaystyle u(0)=\int_0^1(u'v_0'+uv_0).

Show that {v_0} is the solution of some differential equation with appropriate boundary conditions and compute {v_0} explicitly.

H. Brezis, Functional Analysis, Ex 8.18

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A lemma of J. L. Lions

October 13, 2011 Leave a comment

Let X,Y and Z be three Banach spaces with norms \|\cdot \|_X,\ \|\cdot \|_Y and \|\cdot \|_Z. Assume that X \subset Y with compact injection and that Y\subset Z with continuous injection. Prove that

\forall \varepsilon >0 \exists C_\varepsilon \geq 0 satisfying \|u\|_Y \leq \varepsilon \|u\|_X+C_\varepsilon \|u\|_Z,\ \forall u \in X.

Applications:

  1. Prove that for every \varepsilon >0 there exists C_\varepsilon \geq 0 satisfying\displaystyle \max_{[0,1]}|u| \leq \varepsilon \max_{[0,1]}|u^\prime|+C_\varepsilon\|u\|L^1,\ \forall u \in C^1([0,1]).
  2. Pick p>1. Prove that for every \varepsilon >0 there exists C=C(\varepsilon,p) such that \|u\|_{L^\infty(0,1)} \leq \varepsilon \|u^\prime\|_{L^p(0,1)}+C\|u\|_{L^1(0,1)},\ \forall u \in W^{1,p}(0,1).

Source: Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2011

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Sobolev space impossible extension

October 7, 2011 Leave a comment

Let \Omega=\{ (x,y) \in \Bbb{R}^2 : x \in (0,1),\ y \in (0,x^2)\}. And for \beta < 3/2 (!!! Correction: 1<\beta<3/2) consider v: \Omega \to \Bbb{R},\ v(x,y)=x^{1-\beta}. Prove that:

1) \Omega does not have Lipschitz boundary (i.e. its boundary is not locally the graph of Lipschitz functions).

2) v \in H^1(\Omega).

3) For every ball B which contains \Omega, there is no function in H^1(B) which extends v.

This problem gives a counter example which states that if \Omega doesn’t have Lipschitz boundary, then there may be no extension to some functions in H^1(\Omega) to greater Sobolev spaces.

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Generalized Poincare Inequality

October 5, 2011 Leave a comment

Consider \Omega \subset \Bbb{R}^N a bounded domain with Lipschitz boundary. If H is a non-zero, closed subspace in H^1(\Omega), which does not contain the non-zero constant functions, then there is a constant C>0, depending on \Omega, such that \|u \|_{L^2}\leq C \| |\nabla u | \|_{L^2},\ \forall u \in H.

Note that this generalizes the usual Poincare inequality, which says that the above inequality holds for some C>0 on the space H_0^1(\Omega), a space which does not contain the non-zero constant functions.

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Function which vanishes at infinity

Let u \in H^s(\Bbb{R}^n) with s > n/2. Show that \lim_{|x| \to \infty} u(x)=0,

PHD Cincinnati (6103)

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