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- W2037153313 abstract "In general, the value function associated with an exit time problem is a discontinuous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the Hamilton--Jacobi equation involving the proximal subdifferentials (superdifferentials) with subdifferential-type (superdifferential-type) mixed boundary condition. We also show that if the value function is upper semicontinuous, then it is the maximum subsolution of the Hamilton--Jacobi equation involving the proximal superdifferentials with the natural boundary condition, and if the value function is lower semicontinuous, then it is the minimum solution of the Hamilton--Jacobi equation involving the proximal subdifferentials with a natural boundary condition. Futhermore, if a compatibility condition is satisfied, then the value function is the unique lower semicontinuous solution of the Hamilton--Jacobi equation with a natural boundary condition and a subdifferential type boundary condition. Some conditions ensuring lower semicontinuity of the value functions are also given." @default.
- W2037153313 created "2016-06-24" @default.
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- W2037153313 date "2000-01-01" @default.
- W2037153313 modified "2023-09-27" @default.
- W2037153313 title "Discontinuous Solutions of the Hamilton--Jacobi Equation for Exit Time Problems" @default.
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- W2037153313 doi "https://doi.org/10.1137/s0363012997326234" @default.
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