1,455 research outputs found
Direct and Inverse Variational Problems on Time Scales: A Survey
We deal with direct and inverse problems of the calculus of variations on
arbitrary time scales. Firstly, using the Euler-Lagrange equation and the
strengthened Legendre condition, we give a general form for a variational
functional to attain a local minimum at a given point of the vector space.
Furthermore, we provide a necessary condition for a dynamic
integro-differential equation to be an Euler-Lagrange equation (Helmholtz's
problem of the calculus of variations on time scales). New and interesting
results for the discrete and quantum settings are obtained as particular cases.
Finally, we consider very general problems of the calculus of variations given
by the composition of a certain scalar function with delta and nabla integrals
of a vector valued field.Comment: This is a preprint of a paper whose final and definite form will be
published in the Springer Volume 'Modeling, Dynamics, Optimization and
Bioeconomics II', Edited by A. A. Pinto and D. Zilberman (Eds.), Springer
Proceedings in Mathematics & Statistics. Submitted 03/Sept/2014; Accepted,
after a revision, 19/Jan/201
General existence and uniqueness of viscosity solutions for impulse control of jump-diffusions
General theorems for existence and uniqueness of viscosity solutions for
Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVI) with integral
term are established. Such nonlinear partial integro-differential equations
(PIDE) arise in the study of combined impulse and stochastic control for
jump-diffusion processes. The HJBQVI consists of an HJB part (for stochastic
control) combined with a nonlocal impulse intervention term.
Existence results are proved via stochastic means, whereas our uniqueness
(comparison) results adapt techniques from viscosity solution theory. This
paper is to our knowledge the first treating rigorously impulse control for
jump-diffusion processes in a general viscosity solution framework; the jump
part may have infinite activity. In the proofs, no prior continuity of the
value function is assumed, quadratic costs are allowed, and elliptic and
parabolic results are presented for solutions possibly unbounded at infinity
Well-posedness of a nonlinear integro-differential problem and its rearranged formulation
We study the existence and uniqueness of solutions of a nonlinear
integro-differential problem which we reformulate introducing the notion of the
decreasing rearrangement of the solution. A dimensional reduction of the
problem is obtained and a detailed analysis of the properties of the solutions
of the model is provided. Finally, a fast numerical method is devised and
implemented to show the performance of the model when typical image processing
tasks such as filtering and segmentation are performed.Comment: Final version. To appear in Nolinear Analysis Real World Applications
(2016
Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations
We deal with a class of equations driven by nonlocal, possibly degenerate,
integro-differential operators of differentiability order and
summability growth , whose model is the fractional -Laplacian with
measurable coefficients. We state and prove several results for the
corresponding weak supersolutions, as comparison principles, a priori bounds,
lower semicontinuity, and many others. We then discuss the good definition of
-superharmonic functions, by also proving some related properties. We
finally introduce the nonlocal counterpart of the celebrated Perron method in
nonlinear Potential Theory.Comment: To appear in Math. An
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