Coercivity and energy estimates

For the homogeneous Dirichlet equation minus divergence of A gradient u equals F on a bounded Lipschitz domain, assume A is real symmetric, bounded measurable, and uniformly elliptic. Testing with u gives a lower bound lambda times the squared gradient norm. Duality bounds the source pairing by its H-minus-one norm times the gradient norm. After dividing when the gradient is nonzero, the solution norm is bounded by the source norm divided by lambda; the zero-gradient case already satisfies the estimate.

When the source is an L-two function, Cauchy–Schwarz and Poincaré replace its dual norm by the L-two norm times a domain constant. That estimate is useful without knowing the solution explicitly. It also applies to perturbations: the difference of solutions with the same operator and boundary data solves an equation driven by the source difference. A stability constant therefore turns source uncertainty into a solution bound, provided the operator and norm convention are unchanged.

For a potential c bounded below by minus kappa, the form is at least lambda times the gradient energy minus kappa times the L-two energy. If the Poincaré inequality has constant C_P, the remaining gradient coefficient is lambda minus kappa C_P squared. A strictly positive remainder proves coercivity. This is a sufficient condition based on the chosen bounds; its failure alone does not prove that the boundary problem is unsolvable. It means that this particular coercivity argument has stopped providing control.

Bounded first-order terms can be treated with Cauchy–Schwarz and Young's inequality, paying a small part of the gradient energy and leaving an L-two remainder. This produces a Gårding-type lower bound: the form plus a sufficiently large multiple of the L-two norm squared is coercive. The shift changes the operator and gives an invertible reference problem; it does not assert that the unshifted operator has no kernel. Fredholm theory later identifies the remaining finite-dimensional compatibility obstruction.

An interior estimate uses a compactly supported cutoff eta and tests a homogeneous equation with eta squared times u. The resulting cross term is controlled by Young's inequality, leaving the gradient energy on the smaller region bounded by the squared solution times the cutoff gradient on the surrounding region. This Caccioppoli mechanism needs only ellipticity and bounded coefficients for scalar divergence-form weak solutions. It avoids prescribing boundary data because the test function vanishes before reaching the outer boundary.

A typical concentric-ball estimate bounds the gradient L-two norm on radius r by a constant times the solution L-two norm on radius R divided by R−r, for r<R and the larger closed ball inside the domain. The constant depends on dimension and ellipticity bounds. The gap between the balls is essential: localization has a derivative cost. Such a local estimate is different from global uniqueness, and it cannot remove a constant-mode ambiguity in a pure Neumann problem without a normalization.