Maximum principles and comparison

Let Omega be bounded Lipschitz, A bounded measurable, real symmetric and uniformly elliptic, and c bounded nonnegative. Suppose u is H-one, its boundary trace is nonpositive, and its weak inequality a(u,v)≤0 holds for every nonnegative H-zero-one test. The positive part u-plus belongs to H-zero-one. Testing with that function gives a nonnegative gradient term and a nonnegative potential term whose sum is at most zero. Poincaré then forces the positive part to vanish, proving u≤0 almost everywhere.

To obtain an upper boundary bound M, apply this reasoning to u−M. For c≥0, a nonnegative constant M contributes cM≥0, so a subsolution lies below the larger of zero and its boundary supremum. When c=0 the nonnegative restriction on M is unnecessary, and the boundary supremum alone works. These weak statements concern almost-everywhere representatives and traces; a pointwise classical maximum statement additionally assumes enough continuity to evaluate the solution and boundary values at individual points.

If Lu≤Lv weakly and u≤v on the boundary, their difference satisfies the preceding zero-boundary-order inequality and is nonpositive. This proves comparison for the same linear operator. A lower or upper barrier is a function whose operator value and boundary values bound those of the desired solution in the appropriate direction. Barriers give explicit estimates even when the source varies spatially or an exact representation would be inconvenient. They must be checked in the original equation, including every zero-order term.

The strong maximum principle contains more information and needs a compatible theorem. For a classical harmonic function on a connected open domain, an attained interior maximum or minimum forces the function to be constant. One proof uses the mean-value property to propagate equality. The weak maximum principle alone does not establish that propagation, and neither statement says a solution of a positively forced Poisson equation cannot have an interior maximum. The sign of its Laplacian decides which extremum is ruled out.

A negative potential can invalidate the simple maximum principle even though the principal operator remains uniformly elliptic. On the interval from zero to pi, sin x is positive inside, vanishes at the endpoints, and solves minus u-double-prime minus u equals zero. Thus a nonpositive boundary and homogeneous equation do not force a nonpositive solution for that operator. Coercivity or an appropriate positive principal-eigenvalue condition can support other maximum-principle variants, but the missing sign cannot simply be ignored.

Boundary regularity matters differently for different conclusions. The energy proof uses traces and does not need a pointwise normal derivative. A Hopf boundary conclusion about a strict outward derivative requires additional geometric and regularity conditions, such as a suitable interior sphere condition in standard classical statements. We are not deriving that conclusion from a weak trace argument. Keeping the weak, strong, and boundary forms separate makes comparison a reliable tool rather than an all-purpose claim about every elliptic-looking equation.