Definition of weak solutions from geometrical point of view

Question

Why are weak solutions defined like:

A function $u \in H^1(\Omega)$ is a weak solution of $$ Lu:=\operatorname{div}(A\nabla u)+b\cdot\nabla u+cu=f+\operatorname{div}F, \;\text{in } \Omega $$ if $$ \int_\Omega \nabla \phi\cdot (A\nabla u-F)dx=\int_\Omega\phi(b\cdot \nabla u+cu-f) dx $$ holds for every $\phi \in C_0^\infty(\Omega).$

I mean, we define this new notion "weak solutions" in order to generalize classical solutions. Then we just need one purpose which is "classical solution $\Rightarrow$ weak solution" and if everything is sufficiently smooth, then it follows that a weak solution is automatically a classical one. If this is our motivation to define weak solutions, then there would have tons of methods to do.

My point is why this type of definition of weak solutions is so important. Is there any other motivations like from geometry or other subjects to make this definition so outstanding and is there any other type of definitions of generalized solutions?

Answer 1

Generally speaking, you find why a definition is important from seeing it used. Also, any good PDE textbook should point out what is gained by considering weak solutions. I'll give two reasons.

1. The search for classical solutions of PDE of $k$th order is confined to the space $C^k$. This space is not reflexive, and weak compactness arguments are not applicable. In contrast, weak solutions are found in Sobolev spaces, which are typically reflexive (some are even Hilbert spaces, like $H^1$).
2. Sometimes it's not even the PDE that we care about the most, but a variational principle of some sort. The simplest example is the Dirichlet principle: minimization of the energy functional $E(u)=\int |\nabla u|^2$ models a few things in electrostatics and continuum mechanics. Naturally, we seek to minimize $E(u)$ in the space of functions such that $E(u)<\infty$. This brings us to $H^1$ again. Loosely speaking, at the point of minimum the derivative $E'$ should be $0$. When you write down what this means: $$\lim_{h\to 0} \frac{1}{h}(E(u+h\phi)-E(u))=0,\quad \phi\in C_0^\infty$$ and do the computation, you'll see that $u$ is a weak solution of $\operatorname{div}\nabla u=0$. (One could also take $\phi\in C^2_0$ or $\phi\in H_0^1$ here and get the same notion of solution; there is some flexibility in choosing the space of test functions.) From this viewpoint, the notion of weak solution arises naturally, not as an attempt to generalize something, but as a way to record that $u$ is a stationary point of $E$.

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That said, weak solutions have their limitations. They work well for linear and quasilinear equations, but not so for fully nonlinear equations like $u_{xx}u_{yy}-u_{xy}^2=f$. Viscosity solutions which Anthony Carapetis mentioned, help to deal with some nonlinear equations. And for some PDE the notion of a solution is devised ad hoc.

Answer 2

Not a full answer, just some suggestions.

I want to know whether there is some explanation from geometric view.

The weak formulation's well-posedness (solution exists, unique) comes from Lax-Milgram lemma. L-M is generalized by Nečas, which is essentially rephrasing closed range theorem and open mapping theorem in the Hilbertian setting. Any geometrical interpretation of this? I would like to know as well.

In Lax-Milgram, the coercivity is exclusive to Hilbert space. The setting $$ E(v) = \frac{1}{2}a(v,v)-\langle f,v\rangle,\tag{1} $$ translates the coercivity of the bilinear form $a(\cdot,\cdot)$ to the convexity of the functional $E(\cdot)$, so that the minimizer exists.

The weak formulation can be obtained from the first variation of (1) being zero. From physics point of view, the functional represents the energy, for example, in elasticity, the deformation energy in equilibrium is $a(u,u)/2$, and the test function $v$ is a perturbation that does not change the boundary behavior of $u$ (Dirichlet or Neumann). While the first variation of (1) being zero for any $v$: $$ \lim_{\epsilon \to 0}\frac{d}{d\epsilon} E(u+\epsilon v) = 0 $$ is actually the Gâteaux derivative of $E$ in the direction of this perturbation.

Also notice that Riesz representation theorem essentially characterizes the element in the dual space (right hand side of your equation) with the solution $u$ in the original space, again in the Hilbertian setting (inner product). Another question to ask is perhaps: "What is the geometric interpretation of Riesz?"