Obstacle problem
The problem is to find the equilibrium position of an elastic membrane whose boundary is held fixed, and which is constrained to lie above a given obstacle.
It is deeply related to the study of minimal surfaces and the capacity of a set in potential theory as well.
Applications include the study of fluid filtration in porous media, constrained heating, elasto-plasticity, optimal control, and financial mathematics.
[1] The mathematical formulation of the problem is to seek minimizers of the Dirichlet energy functional, in some domains
In general, the solution is continuous and possesses Lipschitz continuous first derivatives, but that the solution is generally discontinuous in the second derivatives across the free boundary.
The free boundary is characterized as a Hölder continuous surface except at certain singular points, which reside on a smooth manifold.
Qualche tempo dopo Stampacchia, partendo sempre dalla sua disequazione variazionale, aperse un nuovo campo di ricerche che si rivelò importante e fecondo.
Si tratta di quello che oggi è chiamato il problema dell'ostacolo.
[2]The obstacle problem arises when one considers the shape taken by a soap film in a domain whose boundary position is fixed (see Plateau's problem), with the added constraint that the membrane is constrained to lie above some obstacle
[3] In this case, the energy functional to be minimized is the surface area integral, or This problem can be linearized in the case of small perturbations by expanding the energy functional in terms of its Taylor series and taking the first term only, in which case the energy to be minimized is the standard Dirichlet energy The obstacle problem also arises in control theory, specifically the question of finding the optimal stopping time for a stochastic process with payoff function
of the obstacle problem can be characterized as the expected value of the payoff, starting the process at
[4] Suppose the following data is given: Then consider the set which is a closed convex subset of the Sobolev space
whose weak first derivatives is square integrable, containing those functions with the desired boundary conditions and whose values above the obstacle's.
; in symbols The existence and uniqueness of such a minimizer is assured by considerations of Hilbert space theory.
is the ordinary scalar product in the finite-dimensional real vector space
This is a special case of the more general form for variational inequalities on Hilbert spaces, whose solutions are functions
of the overall space, such that for coercive, real-valued, bounded bilinear forms
[6] A variational argument shows that, away from the contact set, the solution to the obstacle problem is harmonic.
A similar argument which restricts itself to variations that are positive shows that the solution is superharmonic on the contact set.
Subject to a degeneracy condition, level sets of the difference between the solution and the obstacle,
except on a set of singular points, which are themselves either isolated or locally contained on a
[9] The theory of the obstacle problem is extended to other divergence form uniformly elliptic operators,[6] and their associated energy functionals.
The parabolic, time-dependent cases of the obstacle problem and its variants are also objects of study.
