Ngô Quốc Anh

November 17, 2013

Existence result for quasilinear second order elliptic PDEs via fixed point theorems

Filed under: PDEs — Ngô Quốc Anh @ 0:33

Today, we show how fixed point theorems can be used to obtain some existence results for elliptic PDEs. This topic is adapted from Chapter 9 in Evan’s book. The fixed point theorem that we are going to use is the so-called Schaefer Fixed Point Theorem.

Theorem (Schaefer Fixed Point Theorem). Let X be a real Banach space. Suppose A : X\to X is a continuous and compact mapping. Assume further that the set

\{u \in X:u=\lambda A(u) \quad \text{ for some } 0 \leqslant \lambda \leqslant 1\}

is bounded (in X). Then A has a fixed point.

Following is the PDE that we are going to demonstrate.

\left\{ \begin{array}{rcl} - \Delta u + \mu u &=& f(\nabla u) \quad \text{ in } \Omega , \hfill \\ u &=& 0 \quad \text{ on }\partial \Omega , \hfill \\ \end{array} \right.

where \Omega \subset \mathbb R^n is bounded with smooth boundary \partial \Omega and f : \mathbb R^n \to \mathbb R is a smooth Liptschitz continuous function satisfying the following growth condition

|f(\vec p)| \leqslant C(|\vec p|+1)

for some constant C>0 and all \vec p \in \mathbb R^n. We claim that

Theorem. If \mu>0 is large enough, there exists a function u\in H_0^1(\Omega) \cap H^2(\Omega) solving the above PDE.

To prove this result, we do as follows.

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November 12, 2013

A Neumann boundary inequality by P.L. Lions

Filed under: PDEs — Ngô Quốc Anh @ 6:30

Today, we prove a very interesting inequality along the boundary of a convex set. This proof is adapted from a book by P.L. Lions entitled Generalized Solutions of Hamilton-Jacobi Equations in the series Research Notes In Mathematics Series by Pitman Publishing.

Theorem. Assume that \Omega \subset \mathbb R^n is convex domain and u \in C^2(\overline \Omega). \nu is the outward unit normal vector field along the boundary \partial \Omega. We assume further that the function u satisfies

\displaystyle \frac{\partial u}{\partial \nu}=0 \quad \text{ on }\partial\Omega.

Then we have the following inequality

\displaystyle \frac{\partial }{\partial \nu}(|\nabla u|^2) \leqslant 0 \quad \text{ on }\partial\Omega.

Proof. The proof is surprisingly simple. First, we can assume that \Omega is a sublevel of some convex function, say \phi, such that \phi \in C^2(\mathbb R^n) and |\nabla \phi| \ne 0 along \partial \Omega, i.e.

\Omega = \{x \in \mathbb R^n : \phi(x) < 0\}.

Then the outward unit normal vector field \nu along the boundary \partial \Omega can be calculated using

\displaystyle \nu = \frac{\nabla \phi}{|\nabla \phi|} \quad \text{ along }\partial \Omega.

Note that this is clear since the boundary \partial \Omega can be consider as the level set of \phi at the level 0. Therefore, the gradient \nabla \phi is automatically perpendicular to the level set (and in this case, just the boundary \partial \Omega). (See a similar argument I claimed in this topic about the Pohozaev equality.)

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November 6, 2013

The prescribed scalar curvature: An inf x sup inequality in the positive Yamabe invariant

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 0:56

Today, we talk about an inequality of the form \inf \times \sup \geqslant c for solutions of the so-called prescribed scalar curvature problem, i.e.

\displaystyle -\frac{4(n-1)}{n-2}\Delta_g u + \text{Scal}_g u = V u^{2^\star -1}

where the scalar curvature \text{Scal}_g>0 is fixed but the function V which satisfies

0<a \leqslant V(x) \leqslant b

for any x \in M. Here 2^\star = 2n/(n-2) is the Sobolev critical exponent and n \geqslant 3 is the dimension of M. In addition, the manifold M is compact and has no boundary.

Theorem (Bahoura). There exists a positive constant c depending on a,b,M such that

\displaystyle \sup_M u \times \inf_M u \geqslant c

for any solution u of the PDE.

We prove this theorem using contradiction: There exists a sequence of solutions u_i of

\displaystyle -\frac{4(n-1)}{n-2}\Delta_g u_i + \text{Scal}_g u_i = V_i u_i^{2^\star -1}

such that

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November 3, 2013

The Yamabe problem: The case of spheres

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 23:54

When the manifold (M,g) is the standard sphere (\mathbb S^n, g_{\rm car}), the Yamabe is interesting. First, we cover the so-called Kazdan-Warner obstruction. The original statement is for the prescribing scalar curvature problem, i.e. the following PDE

\displaystyle -\frac{4(n-1)}{n-2}\Delta_g \phi + R\phi = R' \phi^{(n+2)/(n-2)}

where R and R' are scalar curvatures of the metrics g and g'=\phi^{4/(n-2)}g respectively. We note that, instead of using the condition \mu_{2n/(n-2)} = n(n-1)\omega_n^{2/n} to avoid the triviality of the limiting solution, in this generalized equation, one has to use

\displaystyle \mu_{2n/(n-2)} < n(n-1)\omega_n^{2/n} (\sup_M R')^{-(n-2)/n}.

Theorem (Kazdan-Warner obstruction). If \phi >0 is a solution of the preceding PDE on (\mathbb S^n, g_{\rm car}) with n\geqslant 0, then

\displaystyle \int_{\mathbb S^n} \phi^{2n/(n-2)}\langle \nabla^i R' , \nabla_i F \rangle dv_g = 0

for any spherical harmonics F of degree 1.

When n=2, such an obstruction has been mentioned once here. We shall not provide any proof for this obstruction here, it is similar to that for the case n=2 and basically is integration by parts. We note that all spherical harmonics F of degree 1 satisfy

\displaystyle \nabla_{ij} F = -\frac{\lambda_1 F}{n} g_{ij} =: -\alpha^2 Fg_{ij}.

Interestingly, on the sphere (\mathbb S^n, g) having unit volume, constants appearing in the Sobolev inequality are optimal

\displaystyle \|\phi\|_{2n/(n-2)}^2 \leqslant K(n,2)^2 \|\nabla\phi\|_2^2 + \|\phi\|_2^2.

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November 2, 2013

The Yamabe problem: The work by Richard Melvin Schoen

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 3:52

By using the notations used in the note about the work of Yamabe, they are

\displaystyle {F_q}(u) = \frac{{\displaystyle\int_M {\left( {\frac{{4(n - 1)}}{{n - 2}}|\nabla u{|^2} + R{u^2}} \right)d{v_g}} }}{{{{\left( {\displaystyle\int_M {|u{|^q}d{v_g}} } \right)}^{2/q}}}}

where q \leqslant \frac{2n}{n-2} and

\displaystyle {\mu _q} = \mathop {\inf }\limits_{u \in {H^1}(M)} {F_q}(u),

For simplicity, we set

\displaystyle E(u) = \int_M {\left( {|\nabla u{|^2} + \frac{{n - 2}}{{4(n - 1)}}R{u^2}} \right)d{v_g}} .

Schoen proved that:

Theorem. In any case, there holds

\displaystyle\mu_{2n/(n-2)} \leqslant n(n-1)\omega_n^{2/n},

and the equality occurs if and only if M is conformally equivalent to the sphere with standard metric.

In order to prove the above result, it suffices to consider the case either n=3,4,5 or M is locally conformally flat at some point. The main ingredient of his proof is the positive mass theorem. This well-known result says that if the metric g of M is conformally flat in a neighborhood of O, then the Green fucntion of the conformal Laplacian

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November 1, 2013

The Yamabe problem: The work by Thierry Aubin

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 22:05

Following the previous note about the work of Trudinger, today we talk about the work of Aubin regarding to the Yamabe problem, that is the following simple PDE

\displaystyle -\Delta \varphi + R\varphi = C_0 \varphi^\frac{n+2}{n-2}.

In his elegant paper entitlde “Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire” published in J. Math. Pures Appl. in 1976, Aubin proved the existence for almost all manifolds for n\geqslant 6.

By using the notations used in the note about the work of Yamabe, they are

\displaystyle {F_q}(u) = \frac{{\displaystyle\int_M {\left( {\frac{{4(n - 1)}}{{n - 2}}|\nabla u{|^2} + R{u^2}} \right)d{v_g}} }}{{{{\left( {\displaystyle\int_M {|u{|^q}d{v_g}} } \right)}^{2/q}}}}

where q \leqslant \frac{2n}{n-2} and

\displaystyle {\mu _q} = \mathop {\inf }\limits_{u \in {H^1}(M)} {F_q}(u),

Aubin proved that

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