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Characterizations of Sobolev Spaces


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)}.

Proof: (i) {\Rightarrow} (ii). Obvious. Write the definition and apply Holder’s inequality.

(ii) {\Rightarrow} (i). The functional

\displaystyle \varphi \mapsto \int_\Omega u \frac{\partial \varphi}{\partial x_i}

is a continuous linear functional on a dense subspace of {L^{p'}(\Omega)}, and therefore can be extended to a continuous linear functional on the whole {L^{p'}(\Omega)}. By Riesz’s representation theorem there exists a function {g_i \in L^p(\Omega)} such that

\displaystyle \int_\Omega u \frac{\partial \varphi}{\partial x_i}=-\int_\Omega g_i\varphi.

Since this happens for every {i=1..N} we find that {u \in W^{1,p}(\Omega)}.

(i) {\Rightarrow} (iii). Assume first that { u \in C_c^\infty(\Bbb{R}^N)}. Let {h \in \Bbb{R}^N} and set

\displaystyle v(t)=u(x+th),\ t \in \Bbb{R}.

Then we find that {v'(t)=h \cdot \nabla u(x+th)} and therefore

\displaystyle u(x+h)-u(x)=v(1)-v(0)=\int_0^1 v'(t)dt =\int_0^1 h \cdot \nabla u(x+th)dt.

For {1 < p< \leq \infty} we have

\displaystyle |\tau_hu(x)-x)|^p\leq |h|^p \int_0^1 |\nabla u (x+th)|^p dt

and

\displaystyle \int_\omega |\tau_h u(x) -u(x)|^pdx \leq |h|^p \int_\omega\int_0^1 |\nabla u(x+th)|^pdtdx

\displaystyle = |h|^p \int_0^1 \int_{\omega+th} |\nabla u(y)|^pdydt.

If {|h|} is small enough then there exists an open set {\omega' \subset \subset \Omega} such that {\omega +th \subset \omega'} for all {t \in [0,1]} and therefore

\displaystyle \|\tau_hu-u\|_{L^p(\omega)} \leq |h| \|\nabla u\|_{L^p(\omega')},

which concludes the proof for {u \in C_c(\Bbb{R}^N)} and {1 \leq p <\infty}. But we know that {C_c(\Bbb{R}^N)} is dense in {W^{1,p}} in the sense that there exists {u_n \in C_c(\Bbb{R}^N)} such that {u_n \rightarrow u} in {L^p(\Omega)} and {\nabla u_n \rightarrow \nabla u} in {L^p(\omega)} for every {\omega \subset \subset \Omega}. This finishes the argument. For {p=\infty} apply the above for {p<\infty} and make {p \rightarrow \infty}.

(iii) {\Rightarrow} (ii). Let {\varphi \in C_c^\infty(\Omega)} and consider {\omega \subset \subset \Omega} such that {\text{supp} \varphi \subset \omega}. Take {h} with {|h|} small enough such that {\omega+h\subset \subset \Omega}. Because of (iii) we have

\displaystyle \left| \int_\Omega (\tau_hu-u)\varphi \right| \leq C |h| \|\varphi\|_{L^{p'}(\Omega)}.

which translates to

\displaystyle \left|\int_\Omega u(y) \frac{\varphi(y-h)-\varphi(y)}{|h|}dy \right| \leq C \|\varphi\|_{L^{p'}(\Omega)}.

If we choose {h=te_i} in the last inequality and make { t \rightarrow 0} then we obtain (i).

In the case {p=1} the following implications still hold:

\displaystyle \text{(i)} \Rightarrow \text{(ii)} \Leftrightarrow \text{(iii)}.

The functions which satisfy (ii) and (iii) in the case when {p=1} are the functions of bounded variation.

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  1. October 27, 2012 at 6:36 pm

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