divergence form
There are two ways to write the same elliptic operator, and they look almost identical but behave very differently when the coefficients are rough. Divergence form writes the operator as the divergence of a flux: L u = - div(A(x) grad u) = - sum over i,j of the partial in x_i of (a_ij(x) u_(x_j)). It mirrors the physics directly — A grad u is a flux (of heat, charge, or fluid), and taking its divergence and setting the result to a source is a conservation law: what flows in must flow out.
Contrast it with non-divergence form, - sum a_ij u_(x_i x_j), where the coefficients sit outside the derivatives. If A is smooth the two forms are interchangeable — just differentiate the product by the product rule. But if A is merely bounded and measurable (jumping across material interfaces, say), you cannot differentiate it, and only the divergence form still makes sense. The trick is that divergence form pairs perfectly with integration by parts: multiply by a test function v, integrate, and the derivative moves onto v, leaving the integral of A grad u dot grad v. That symmetric integral is the bilinear form behind the weak formulation, and it never asks A to be differentiated at all.
This is why divergence form is the home of weak solutions, the energy method, and the whole variational theory: Lax-Milgram, the Dirichlet principle, finite elements. The divergence-form equation - div(A grad u) = f is exactly the Euler-Lagrange equation of the energy integral of one-half A grad u dot grad u minus f u. Non-divergence form, by contrast, is the natural setting for the maximum principle and pointwise estimates. Knowing which form you are in tells you which toolbox to reach for.
Two slabs of different materials are glued together, so the conductivity a(x) jumps from 1 to 5 across the seam. The steady temperature solves - (a(x) u')' = f in divergence form. Multiplying by a test function and integrating by parts gives the integral of a u' v' = the integral of f v — no derivative of a is ever needed, so the jump causes no trouble. The weak solution is continuous and its flux a u' is continuous across the seam, even though u' itself jumps.
Divergence form survives jumping coefficients because integration by parts moves the derivative off A entirely.
Divergence form and non-divergence form give the same equation only when A is differentiable. With merely measurable coefficients they are genuinely different problems with different solution concepts, and the celebrated De Giorgi-Nash-Moser regularity theorem is a statement about the divergence-form case specifically.