v subdifferentials of convex operators abstract for any convex operator from a convex set c of a topological vector space e into another one f endowed with a convex cone f a notion of v subdifferential v a of at a c is introduced although it is equivalent to the notion of e sub differential when f r it enjoys many important properties which are not satisfied by the e subdifferential whenever int f o the nonvacuity of v a is proved whenever v is a neighbourhood of zero in f and belongs to a large class of mappings analogous to the class of lower semicontinuous real valued convex functions other properties of v subdifferentials are studied and applications to differentiability of are made

a deterministic control based approach motion by curvature the level set formulation of motion by mean curvature is a degenerate parabolic equation we show that its solution can be interpreted as the value function of a deterministic two person game more precisely we give a family of discrete time two person games whose value functions converge in the continuous time limit to the solution of the motion by curvature pde for a convex domain the boundary s first arrival time solves a degenerate elliptic equation this corresponds in our game theoretic setting to a minimum exit time problem for a nonconvex domain the two person game still makes sense we draw a connection between its minimum exit time and the evolution of curves with velocity equal to the positive part of the curvature these results are unexpected because the value function of a deterministic control problem is normally the solution of a first order hamilton jacobi equation our situation is different because the usual first order calculation is singular 2005 wiley periodicals inc

infinity laplacian equation with strong absorptions abstract we study regularity properties of solutions to reaction diffusion equations ruled by the infinity laplacian operator we focus our analysis in models presenting plateaus i e regions where a non negative solution vanishes identically we obtain sharp geometric regularity estimates for solutions along the boundary of plateaus sets in particular we show that the n hausdorff measure of the plateaus boundary is finite for a universal number 0

output based event triggered control with performance guarantees we propose an output based event triggered control solution for linear discrete time systems with a performance guarantee relative to periodic time triggered control while reducing the communication load the performance is expressed as an average quadratic cost and the plant is disturbed by gaussian process and measurement noises we establish several connections with previous works in the literature discussing in particular the relation to absolute and relative threshold policies the usefulness of the results is illustrated through a numerical example

event triggered h control for discrete time piecewise affine systems with norm bounded uncertainties this paper is concerned with the problem of designing event triggered h controllers for discrete time piecewise affine pwa systems with norm bounded uncertainties event triggered scheme transmi

a unified approach to duality in convex programming abstract bifunctional duality lagrange duality and fenchel duality in convex programming are presented under a common point of view stability criteria weaker and more natural than the usual slater type constraint qualification hypotheses are obtained for any of these types of duality

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

an optimization problem with volume constraint for a degenerate quasilinear operator abstract we consider the optimization problem of minimizing u p d x with a constraint on the volume of u 0 we consider a penalization problem and we prove that for small values of the penalization parameter the constrained volume is attained in this way we prove that every solution u is locally lipschitz continuous and that the free boundary u 0 is smooth

continuous affine support mappings for convex operators abstract we investigate the problem of existence of continuous affine resp linear support mappings for convex resp sublinear partially defined operators we prove that a weakly borel measurable convex operator defined on a closed convex subset of a locally convex frechet space with values in an ordered locally convex vector space with the least upper bound property and an order unit inducing the topology in fact has continuous affine support mappings we give an example indicating that the order unit being present is essential a step towards our result is provided by a variant of the sandwich theorem obtained for the case of weakly borel measurable convex and concave operators finally we obtain necessary and sufficient conditions for the existence of a continuous affine support mapping h for a convex operator such that h x 0 x 0 holds for a fixed point x 0 in terms of the upper directional derivate x 0 for at x 0

conjugate convex operators convex mappings from a locally convex space x into f f are considered where f is an ordered topological vector space and an arbitrary greatest element adjoined to f in view of applications to the polarity theory of convex operators the possibility is investigated of representing a convex mapping taking values in f as a supremum of continuous affine mappings

a constrained shape optimization problem in orlicz sobolev spaces abstract in this manuscript we study the following optimization problem given a bounded and regular domain r n we look for an optimal shape for the w vanishing window on the boundary with prescribed measure over all admissible profiles in the framework of the orlicz sobolev spaces associated to constant for the sobolev trace embedding in this direction we establish existence of minimizer profiles and optimal sets as well as we obtain further properties for such extremals finally we also place special emphasis on analyzing the corresponding optimization problem involving an a vanishing hole inside the domain with volume constraint

the formation of the dead core in parabolic reaction diffusion problems on donne des conditions pour l existence d une region a concentration de reactant nulle pour des problemes paraboliques non lineaires de reaction diffusion

the infinity laplacian aronsson s equation and their generalizations the infinity laplace equation u 0 arose originally as a sort of euler lagrange equation governing the absolute minimizer for the l variational problem of minimizing the functional ess sup u du the more general functional ess sup u f x u du leads similarly to the so called aronsson equation a f u 0 in this paper we show that these pde operators and various interesting generalizations also appear in several other contexts seemingly quite unrelated to l variational problems including two person game theory with random order of play rapid switching of states in control problems etc the resulting equations can be parabolic and inhomogeneous equation types precluded in conventional l variational problems

on viscosity solutions of fully nonlinear equations with measurable ingredients we study fully nonlinear uniformly elliptic equations with measurable ingredients caffarelli s recent work on w2 p estimates for viscosity solutions has led to significant progress in this area here we present a unified treatment of this theory based on an appropriate notion of viscosity solution for instance it is shown that strong solutions are viscosity solutions that viscosity solutions are twice differentiable a e and that the pointwise derivatives satisfy the equation a e an important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations this extends results of this type due to evans and krylov our work is to some extent expository the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients 1996 john wiley sons inc

suboptimal event triggered control for time delayed linear systems this technical note considers event triggering conditions and controller synthesis approaches for delayed linear systems optimization problems for minimizing the upper bound of quadratic cost functions are formulated in the form of linear matrix inequalities lmis by solving the optimization problems a unique control gain can be obtained the performance considered in this technical note includes a linear quadratic cost function for quantifying the control performance and average event times at which the control input must be updated for quantifying the transmission reductions comparisons with other approaches in the literature are given to demonstrate the advantages with respect to the two performance indices furthermore an experimental implementation of the proposed methods in an inverted pendulum system shows the applicability and effectiveness in real world

reaction diffusion equations for the infinity laplacian abstract we derive sharp regularity for viscosity solutions of an inhomogeneous infinity laplace equation across the free boundary when the right hand side does not change sign and satisfies a certain growth condition we prove geometric regularity estimates for solutions and conclude that once the source term is comparable to a homogeneous function then the free boundary is a porous set and hence has zero lebesgue measure additionally we derive a liouville type theorem when near the origin the right hand side grows not faster than third degree homogeneous function we show that if a non negative viscosity solution vanishes at a point then it has to vanish everywhere

tug of war and the infinity laplacian we consider a class of zero sum two player stochastic games called tug of war and use them to prove that every bounded real valued lipschitz function f on a subset y of a length space x admits a unique absolutely minimal am extension to x i e a unique lipschitz extension u x for which lip u u lip u u for all open u x y

regularity for degenerate evolution equations with strong absorption abstract in this manuscript we study geometric regularity estimates for degenerate parabolic equations of p laplacian type 2 p under a strong absorption condition p u u t 0 u q in t 0 t where 0 q 1 this model is interesting because it yields the formation of dead core sets i e regions where non negative solutions vanish identically we shall prove sharp and improved parabolic c regularity estimates along the set f 0 u t u 0 t the free boundary where p p 1 q 1 1 p 1 some weak geometric and measure theoretical properties as non degeneracy positive density porosity and finite speed of propagation are proved as an application we prove a liouville type result for entire solutions a specific analysis for blow up type solutions will be done as well the results are new even for dead core problems driven by the heat operator

uniqueness of lipschitz extensions minimizing the sup norm of the gradient in this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values geometrically this can be interpreted as finding a minimal lipschitz extension due to the weak convexity of the functional associated to this problem solutions are generally nonunique by adopting g aronsson s notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation in fact we actually prove a weak maximum principle for this partial differential equation which in some sense is the euler equation for the minimization problem this is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity to overcome this difficulty we use the weak solutions of m g crandall and p l lions also known as viscosity solutions in conjunction with some arguments using integration by parts

regularity for quasilinear equations on degenerate singular sets we prove a new universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at singular set of degeneracy fancyscript s u x d u x 0 our main theorem reveals that along fancyscript s u u is asymptotically as regular as solutions to constant coefficient equations in particular along the critical set fancyscript s u du enjoys a modulus of continuity much superior than the possibly low continuity feature of the coefficients the results are new even in the context of linear elliptic equations where it is herein shown that h 1 weak solutions to text div left a ij x du right 0 with a ij elliptic and dini continuous are actually c 1 1 along fancyscript s u the results and insights of this work foster a new understanding on smoothness properties of solutions to degenerate or singular equations beyond typical elliptic regularity estimates precisely where the diffusion attributes of the equation collapse

generic gâteaux differentiability of convex functions on small sets the theory of generic differentiability of convex functions on banach spaces is by now a well explored part of infinite dimensional geometry all the attempts to solve this kind of problem have in common as a working hypothesis one special feature of the finite dimensional case namely convex functions are always considered to be defined on convex sets with nonempty interior but typically a convex set in a banach space does not have interior points even when it is not contained in a closed hyperplane so this raises the problem of expounding a theory of differentiability of convex functions defined on small sets i e sets without interior points in our paper n2 we have made an attempt to solve this problem discussing questions of generic frechet differentiability of convex functions on small sets in the present paper we deal with problems of generic gateauxdifferentiability in this context

universal moduli of continuity for solutions to fully nonlinear elliptic equations this paper provides universal optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations f x d2u f x based on the weakest and borderline integrability properties of the source function f in different scenarios the primary result established in this work is a sharp log lipschitz estimate on u based on the ln norm of f which corresponds to optimal regularity bounds for the critical threshold case optimal c1 regularity estimates are also delivered when f in l n varepsilon the limiting upper borderline case f in l infty also has transcendental importance to elliptic regularity theory and its applications in this paper we show under the convexity assumption on f that u in c 1 rm log lip provided f has bounded mean oscillation once more such an estimate is optimal for the lower borderline integrability condition allowed by the theory we establish interior a priori estimates on the c 0 frac n 2 varepsilon n varepsilon norm of u based on the ln e norm of f where is the escauriaza universal constant the exponent frac n 2 varepsilon n varepsilon is optimal when the source function f lies in lq n q n e we also obtain the exact improved sharp holder exponent of continuity

sandwich theorem and supportability for partially defined convex operators abstract given a convex operator e f and a concave operator e f both defined an a locally convex frechet space and having values in an ordered locally convex vector space f with the least upper bound property and satisfying we ask for conditions ensuring the existence of a continuous affine mapping h e f satisfying h besides this continuous sandwich problem we discuss the problem of existence of continuous linear support mappings for partially defined sublinear operators by reducing it to a sandwich problem

nonlocal tug of war and the infinity fractional laplacian motivated by the tug of war game studied by peres et al in 2009 we consider a nonlocal version of the game that goes as follows at every step two players pick respectively a direction and then instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount e 0 as is done in the classical case it is an s stable levy process that chooses at the same time both the direction and the distance to travel starting from this game we heuristically derive a deterministic nonlocal integrodifferential equation that we call the infinity fractional laplacian we study existence uniqueness and regularity both for the dirichlet problem and for a double obstacle problem both problems having a natural interpretation as tug of war games 2011 wiley periodicals inc

an optimization problem with volume constraint with applications to optimal mass transport in this manuscript we study the following optimization problem with volume constraint min left frac 1 p int omega nabla v pdx int partial omega gv ds colon v in w 1 p left omega right text and v 0 leq alpha right here g is acontinuous function and alpha is a fixed constant such that 0 0 we prove that a minimizer exists and satisfies left begin array ccl delta p u p 0 text in u p 0 cup u p 0 5pt nabla u p p 2 frac partial u p partial nu g text on partial omega cap partial u p 0 cup u p 0 5pt u p 0 alpha end array right next we analyze the limit as p to infty we obtain that any sequence of weak solutions converges up to a subsequence lim limits p j to infty u p j x u infty x uniformly in overline omega and uniform limits u infty are solutions to the maximization problem with volume constraint max left int partial omega gv ds colon v in w 1 infty left omega right nabla v l infty omega leq 1 text and v 0 leq alpha right furthermore we obtain the limit equation that is verified by u infty in the viscosity sense finally it turns out that such a limit variational problem is connected to the monge kantorovich mass transfer problem with the involved measures are supported on partial omega and along the limiting free boundary partial u infty neq 0

an optimization problem with volume constraint we consider the optimization problem of minimizing dirichlet integral int omega nabla u 2 with volume constraint on the set u 0 we use a penalization method which for small values of the penalization parameter leads to a solution of the original problem

regularity for the fully nonlinear dead core problem we establish new geometric regularity estimates for reaction diffusion equations with strong absorption terms the model is given by a fully nonlinear elliptic equation with measurable coefficients and a mu holder continuous convection term f x d 2u f u the lack of lipschitz regularity of the map u mapsto f u allows the existence of plateaus i e nonnegative solutions may vanish identically within an a priori unknown region the dead core of the solution we prove that at any touching ground point z in partial u 0 solutions are varkappa mu differentiable for a sharp value varkappa mu ge 2 and in fact varkappa 1 infty the proof is based on a new flatness improvement method we apply this new regularity estimate to establish a liouville type theorem for entire solutions to dead core problems and also to obtain measure estimates on the touching ground boundary the results obtained in this article are new even for dead core problems ruled by linear equations