{open_quotes}Unbounded{close_quotes} second order partial differential equations in infinite dimensional Hilbert spaces

Journal Article · Sat Dec 31 04:00:00 EST 1994 · Communications in Partial Differential Equations

OSTI ID:146815

Swiech, A ^[1]

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1. Georgia Institute of Technology, Atlanta, GA (United States)

In this paper we study fully nonlinear stationary partial differential equations. Where H is a real, separable Hilbert space, A is a linear, densely defined, maximal monotone operator on H and {mu} is a real valued function. Above Du(x) and D{sup 2}u(x) correspond respectively to the first and second order Frechet derivatives of {mu}. Identifying H with its dual, Du(x) corresponds to an element of H and D{sup 2}{mu}(x) to an element of S(H), the space of bounded, self-adjoint operators on H. Equations and initial value problems are of importance since they appear as the dynamic programming equations associated with problems of stochastic optimal control if one controls an infinite dimensional system governed by a stochastic PDE. We call then {open_quotes}unbounded{close_quotes} because of the term (Ax,Du) which is not defined everywhere and is not locally bounded. This forces us to interpret the equations in the proper viscosity sense, in particular the above mentioned term must be given special treatment. {open_quotes}Unboundedness{close_quotes} in dynamic programming equations arises for instance when state equations of systems involve unbounded operators.

• OSTI ID: 146815
• Journal Information: Communications in Partial Differential Equations, Journal Name: Communications in Partial Differential Equations Journal Issue: 11-12 Vol. 19; ISSN 0360-5302; ISSN CPDIDZ
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
99 GENERAL AND MISCELLANEOUS
ALGORITHMS
ANALYTICAL SOLUTION
CAUCHY PROBLEM
DYNAMICS
HILBERT SPACE
MATHEMATICAL OPERATORS
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS