Weak solutions and Galerkin approximation

Take a bounded Lipschitz domain Ω, a finite time interval, and homogeneous Dirichlet boundary data. Consider uₜ−div(A(x,t)∇u)=f, where A is a bounded measurable real symmetric matrix field satisfying λ|ξ|²≤ξᵀAξ≤Λ|ξ|² for fixed positive λ and Λ. For u₀ in L² and f in L² in time with values in H⁻¹, the natural solution belongs to L² in time with values in H₀¹, and its time derivative belongs to L² with values in H⁻¹. No pointwise second spatial derivative is required by this formulation.

Testing against v in H₀¹ gives a dual pairing for uₜ and an ordinary integral of A∇u dotted with ∇v. The boundary condition is encoded by the zero Sobolev trace, not by assigning pointwise boundary values to an arbitrary equivalence class. The solution has a representative continuous in time with values in L², allowing u(0)=u₀ to be imposed strongly in that norm. Spatial almost-everywhere representatives alone would not make an isolated time slice meaningful; the time regularity is doing essential work.

Choose finite-dimensional spaces spanned by Dirichlet Laplacian eigenfunctions, orthonormal in L² and orthogonal in the gradient inner product. Set uN to a finite combination with time-dependent coefficients and require the weak equation against every basis function. This gives a linear finite-dimensional system of ordinary differential equations with measurable bounded matrix coefficients and integrable forcing. Its initial coefficients are the L² projection of u₀. Using uN itself as test produces the energy estimate because all coefficients are sufficiently regular in time for that calculation.

Uniform ellipticity controls the gradient term from below. The forcing is bounded by its H⁻¹ norm times the gradient norm, using the gradient norm on H₀¹ and its dual norm. Young's inequality absorbs half of the gradient term into the left side. Integrating in time yields bounds on the L∞-in-time L² norm and L²-in-time gradient norm independent of N. The equation also controls the time derivative in the corresponding dual space when the projection is chosen with the needed H₀¹ boundedness, as this eigenfunction construction provides.

Weak compactness supplies a subsequence converging in the energy spaces. Because the PDE is linear and A is fixed bounded data, the weak limits pass through the bilinear form when tested appropriately. Time integration by parts identifies the initial trace. Energy estimates then give uniqueness, so different convergent subsequences cannot describe different weak solutions. Starting with smoother initial approximations and using the same stability estimate extends the construction to L² initial data. The resulting existence statement is at the weak-solution level.

Nonlinear terms require additional care: weak convergence of uN does not generally imply convergence of a nonlinear function of uN. Compactness in a stronger topology or a suitable monotonicity argument may be needed. Likewise, weak existence alone does not justify evaluating a maximum at a point or differentiating the PDE repeatedly. Additional data, coefficient, and boundary regularity support later upgrades. Keeping these proof stages separate prevents a formal integration by parts from silently assuming the very solution regularity the construction is meant to establish.