Ngô Quốc Anh

February 28, 2012

The Rellich embedding theorem on a bounded domain

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 18:18

Today, let us summarize steps during the standard proof of the Rellich embedding theorem on a bounded domain \Omega \subset \mathbb R^n. This theorem says that

Theorem (Rellich). Suppose that \Omega \subset \mathbb R^n is an open, bounded domain with C^1 boundary,  and that 1<p< n. Then W^{1,p}(\Omega) is compactly embedded in L^q(\Omega) for all 1\leqslant q <\frac{np}{n-p}.

In particular, for any sequence \{u_j\}_j \subset W^{1,p}(\Omega) , there exists a subsequence \{u_{j_k}\}_k \subset \{u_j\}_j such that u_{j_k} \to u strongly in L^q(\Omega) for some u \in L^q(\Omega).

In order to prove the Rellich theorem, we need the so-called Arzela-Ascoli theorem.

Lemma (The Arzela-Ascoli theorem). Suppose that u_j \in C^0(\overline\Omega), \|u_j\|_{C^0(\overline\Omega)} \leqslant M < \infty, and \{u_j\}_j is equicontinuous. Then there exists a subsequence u_{j_k} \to u uniformly on \Omega.

The Arzela-Ascoli theorem is well-known. To prove the Rellich theorem, we shall use the standard mollifiers. To do that, we have to extend \Omega a little bit.

Step 1. Assume that \overline\Omega \subset \mathbb R^n is also an open, bounded domain with C^1 boundary with \Omega \subset\subset \overline \Omega. By the Sobolev extension theorem, we can define Eu_j by \overline u_j with

\sup_j \|\overline u_j\|_{W^{1,p}(\mathbb R^n)} < C<\infty.

By the Gagliardo-Nirenberg inequality,

\sup_j \|\overline u_j\|_{L^q(\Omega} < C<\infty.

For \varepsilon>0, let \eta_\varepsilon denote the standard mollifiers and set \overline u_j^\varepsilon = \eta_\varepsilon * \overline u_j. By choosing \varepsilon sufficiently small, \overline u_j^\varepsilon \in C^\infty(\overline \Omega).

Step 2. Since

\displaystyle\overline u _j^\varepsilon = \int_{B(0,\varepsilon )} {\frac{1}{{{\varepsilon ^n}}}\eta \left( {\frac{y}{\varepsilon }} \right){{\overline u }_j}(x - y)dy} = \int_{B(0,1)} {\eta (z){{\overline u }_j}(x - \varepsilon z)dz}

we can get that

\displaystyle |\overline u _j^\varepsilon (x) - {\overline u _j}(x)| = \varepsilon \int_{B(0,1)} {\eta (z)\left( {\int_0^1 {|D{{\overline u }_j}(x - \varepsilon tz)|dt} } \right)dz} .

Therefore,

\displaystyle\int_{\overline \Omega } {|\overline u _j^\varepsilon (x) - {{\overline u }_j}(x)|dx} \leqslant \varepsilon {\left\| {D{{\overline u }_j}} \right\|_{{L^1}(\overline \Omega )}} \leqslant \varepsilon {\left\| {D{{\overline u }_j}} \right\|_{{L^p}(\overline \Omega )}} < C.

This and the L^p-interpolation inequality imply that

\displaystyle {\left\| {\overline u _j^\varepsilon - {{\overline u }_j}} \right\|_{{L^q}(\overline \Omega )}} < C\varepsilon .

Step 3. Our goal is to employ the Arzela-Ascoli theorem. It is not hard to see the following

\displaystyle\mathop {\sup }\limits_j {\left\| {\overline u _j^\varepsilon } \right\|_{{C^0}(\overline \Omega )}} \leqslant {\left\| {{\eta _\varepsilon }} \right\|_{{L^\infty }({\mathbb{R}^n})}}\mathop {\sup }\limits_j {\left\| {{{\overline u }_j}} \right\|_{{L^1}(\overline \Omega )}} \leqslant \frac{C}{{{\varepsilon ^n}}} < \infty

and

\displaystyle\mathop {\sup }\limits_j {\left\| {D\overline u _j^\varepsilon } \right\|_{{C^0}(\overline \Omega )}} \leqslant {\left\| {D{\eta _\varepsilon }} \right\|_{{L^\infty }({\mathbb{R}^n})}}\mathop {\sup }\limits_j {\left\| {{{\overline u }_j}} \right\|_{{L^1}(\overline \Omega )}} \leqslant \frac{C}{{{\varepsilon ^{n + 1}}}} < \infty.

Hence there exists a subsequence u_{j_k} which converges uniformly on \overline \Omega so that

\displaystyle\mathop {\lim \sup }\limits_{k,l \to \infty } {\left\| {\overline u _{{j_k}}^\varepsilon - {{\overline u }_{{j_l}}}} \right\|_{{L^q}(\overline \Omega )}} = 0.

By the triangle inequality, we easily get that

\displaystyle\mathop {\lim \sup }\limits_{k,l \to \infty } {\left\| {\overline u _{{j_k}} - {{\overline u }_{{j_l}}}} \right\|_{{L^q}(\overline \Omega )}} < C \varepsilon .

Step 4. By choosing \varepsilon=\frac{1}{2}, \frac{1}{4},... and using the diagonal argument to extract further subsequences, we can arrange to find a subsequence u_{j_{k_l}} \subset u_{j_k} such that

\displaystyle\mathop {\lim \sup }\limits_{l,m \to \infty } {\left\| {\overline u _{j_{k_l}} - {{\overline u }_{j_{k_m}}}} \right\|_{{L^q}(\overline \Omega )}} < C\varepsilon.

Thus,

\displaystyle\mathop {\lim \sup }\limits_{l,m \to \infty } {\left\| u _{j_{k_l} - u _{j_{k_m}}} \right\|_{{L^q}(\Omega )}} =0.

This concludes the proof.

We know discuss a simple application of the idea of the above proof. Assume that the sequence of functions v_j \in [0,1] solves the following

-\Delta v_j = h_jv_j + f_jv_j^{2^\star} \quad \text{ in } \mathbb R^n,

where f_j and h_j are smooth functions and uniformly bounded in \mathbb R^n. We claim that, up to subsequences, v_j \to v strongly in C^2_{\rm loc}(\mathbb R^n). In other words, for fix R>0, the sequence \{v_j\}_j is always compact in C^2(B_R(0)). To see this, we use the standard elliptic regularity theory.

Step 1. Indeed, since v_j are uniformly bounded in \mathbb R^n, we can see that v_j are uniformly bounded in L_{\rm loc}^p(\mathbb R^n) for any p \geqslant 1. Since all coefficients of the PDE are uniformly bounded (and even smooth) in \mathbb R^n, we get via the standard L^p-estimates that v_j \in W_{\rm loc}^{2,q}(\mathbb R^n) for any q \geqslant 1. Using the Sobolev embedding theorem, the following W_{\rm loc}^{2,q}(\mathbb R^n) \hookrightarrow C_{\rm loc}^{0,\alpha}(\mathbb R^n) is compact for some \alpha \in (0,1). By the Schauder estimates, we can conclude that v_j \in C_{\rm loc}^{2,\alpha}(\mathbb R^n).

Step 2. Since all coefficients of the PDE are bounded, we can conclude that the sequence \{v_j\} is bounded in C_{\rm loc}^{2,\alpha}(\mathbb R^n).

Step 3. To complete the proof, let say we have K a compact subset in \mathbb R^n. It is obvious to see that the embedding C^{0,\alpha}(K) \hookrightarrow C^0(K) is compact. Indeed, the uniformly boundedness and the equicontinuous come from the following estimates

\displaystyle |{D^2}{v_j}(x) - {D^2}{v_j}(y)| = \frac{{|{D^2}{v_j}(x) - {D^2}{v_j}(y)|}}{{|x - y{|^\alpha }}}|x - y{|^\alpha } \leqslant 2 {\rm diam}{(K)^\alpha }{\left\| {{D^2}{v_j}} \right\|_{{C^{0,\alpha }}(K)}}

and

\displaystyle |{D^2}{v_j}(x) - {D^2}{v_j}(y)| \leqslant 2 {\rm diam}{(K)^{\alpha-1} }{\left\| {{D^2}{v_j}} \right\|_{{C^{0,\alpha }}(K)}}|x-y|.

By the Arzela-Ascoli theorem, there is a subsequence D^2v_{j_k} converges uniformly in K. In order to use this compact embedding, we need some boundedness of v_j in C_{\rm loc}^{2,\alpha}(\mathbb R^n). Thanks to Step 2, we can conclude that v_j \to v strongly in C^2_{\rm loc}(\mathbb R^n) up to subsequences.

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