Weak Solutions

1 Classical Solutions

1.1 Classifications

Definition 1.1 (Poisson equations with different boundaries)

Dirichlet boundary:

$$− \Delta u = f \text{ in } \Omega, \\ u = g \text{ on } \partial\Omega$$

Neumann boundary:

$$− \Delta u = f \text{ in } \Omega, \\ \partial u = h \text{ on } \partial\Omega$$

If $f=0$, we call it Laplace equation. Classically we restrict the solution $u \in C^2(\Omega) \cap C^0(\overline\Omega)$.

1.2 Uniqueness and Existence

2 Weak Solutions

2.1 Abstract Formulation

Consider the following problem,

$$− \Delta u = f \text{ in } \Omega, u = 0 \text{ on } \partial\Omega \tag{2.1}$$

Definition 2.1 (Weak solution). We say that $u \in H^1_0(\Omega)$ is a weak solution if

$$\int_{\Omega} \nabla u \cdot \nabla v = \int_{\Omega} fv, ∀v \in H^1_0(\Omega)$$

which is called the weak formulation of (2.1). $f\in L^2(\Omega)$ is enough, but $f$ might also be distribution.

Definition 2.2 (Abstract variational problem).

Let $V$ be a Hilbert space,

$$\text{find } u \in V : a(u, v) = F(v), \forall v \in V$$

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Formulating in the abstract way, let $V=H^1_0 (\Omega)$ , We say that $u \in V$ is a weak solution of (2.1) with $f \in V’$ if

$$a(u, v) = F(v), \forall v \in V.$$

where $a(u, v) = \int_{\Omega} \nabla u \cdot \nabla v, F(v)= \left< f,v\right >$. We can denote $(H^1_0 (\Omega))’$ by $H^{−1 }(\Omega)$.

We can utilize the abstract form with Lax-Milgram theorem in PDEs.

2.2 Applications to PDEs

2.2.1 Homogeneous Poisson equation with Dirichlet boundary.

Lemma 2.1 For any $f \in H^{−1 }(\Omega)$, (2.1) in a bounded domain $\Omega$ has a unique weak solution $u ∈ H^1_0 (\Omega)$ which satisfies $\Vert u\Vert_{H^1_0 (\Omega)} = \Vert f\Vert_{H^{−1}(\Omega)}$.

Proof.

(1) Coercivity of $a$.

Using Poincare inequality,

$$\Vert f\Vert_{L^2(\Omega)} \leq C_P \Vert Df\Vert_{L^2(\Omega)}, ∀f \in H^1_0(\Omega).$$

We can show $\Vert\cdot\Vert_{L^2}$ is equivalent to $\Vert\cdot\Vert_{H^1_0}$. Thus, it is direct to get $\alpha=1$.

2.2.2 Non-homogeneous Poisson equation with Dirichlet boundary.

$$− \Delta u = f \text{ in } \Omega, u = g \text{ on } \partial\Omega \tag{2.2}$$

Lemma 2.2. The space $Rτ \subset L^2 (\partial\Omega)$ a strict subspace, endowed with the norm

$$\Vert g\Vert_{R\tau} = \mathop{inf}\limits_{u\in H^1(\Omega), \tau u=g}\Vert u\Vert _{H^1(\Omega)}.$$

is a Banach space, which is usually denoted by $H^{\frac{1}{2}}(\partial\Omega)$.

Definition 2.3 Let $\Omega \subset \mathbb R^n$ be a bounded domain with $C^1$ boundary. We say that $u \in H^1 (\Omega)$ is a weak solution of (2.2) with $f \in H^{-1}(\Omega)$ and $g \in H^{1/2} (\partial\Omega)$ if it satisfies $\tau u = g$ and

$$\int_{\Omega} \nabla u \cdot \nabla v = \int_{\Omega} fv, \forall v \in H_0^1(\Omega)$$

Let $u=u_0+v_g$ such that $\tau v_g=g$ and $v_g\in H^1(\Omega)$, $u_0\in H^1_0(\Omega)$

$$\int_{\Omega}\nabla u \cdot \nabla v=\int_{\Omega} fv-\int_{\Omega}\nabla v_g \cdot \nabla v, \forall v \in H_0^1(\Omega)$$

Remark. According to Lemma 2.2, it is necessary to have some constraint on $g$.

Lemma 2.3 Let $\Omega \subset \mathbb R^n$ be a bounded domain with $C^1 $ boundary. For any $g \in H^{1/2} (\partial\Omega)$ and $f \in H^{-1} (\Omega)$ there exists a unique weak solution to problem.

2.2.3 Non-homogeneous Generalized Poisson equation with Neumann boundary.

The Problem is

$$− \Delta u +cu= f \text{ in } \Omega, \partial_v u = h \text{ on } \partial\Omega \tag{2.3}$$

Definition 2.4. We say that $u \in H^1 (\Omega)$​ is a weak solution of (2.3) with $f \in L^2 (Ω)$​ and $h \in L^2 (\partial\Omega)$​ if

$$\int_{\Omega}\nabla u \cdot \nabla v+\int_{\Omega} u v=\int_{\Omega} fv+\int_{\partial \Omega} h \tau v,\forall v \in H^1 (\Omega).$$

Remark

(1) Here we must require $c>0$ to ensure coercivity of $a$.

(2) Generally, $f\in H^{-1}(\Omega)$ is enough. Since $\tau$ is also a bounded operator, $h\in H^{-1/2}(\Omega)=(H^{1/2}(\Omega))’$ is also enough.

Then the theorem is similarly stated.