mixed boundary value problems for elliptic and parabolic differential equations of second order on considere les problemes aux valeurs limites mixtes suivants lu f 1 dans mu f 2 sur u f 3 sur avec l operateur elliptique et pu f 1 dans 0 t mu f 2 sur 0 t u f 3 sur 0 t 0 pour des operateurs p paraboliques pour des operateurs obliques generaux m et pour des domaines ou et se rencontrent a des angles arbitraires on montre l existence de solutions classiques quand les angles sont non nuls et f 3 est continue de holder

almost sure and pth moment stability and stabilization of regime switching jump diffusion systems this work focuses on regime switching jump diffusions which include three classes of random processes brownian motions poisson processes and markov chains first a scalar linear system is treated as a benchmark model then stabilization of systems with one sided linear growth is considered next nonlinear systems that have a finite explosion time are treated in which regularization explosion suppression and stabilization are achieved by introducing appropriate diffusions together with poisson and markov chain perturbations this work reveals the impact of various random effects on the underlying systems for almost sure and p th moment stability and provides insight on stability and stabilization of switching jump diffusion systems

global c regularity for second order quasilinear elliptic euqations in divergence form on considere des equations elliptiques quasilineaires d ordre 2 sous forme divergence sous des conditions aux limites de dirihlet

on competitive lotka volterra model in random environments focusing on competitive lotka volterra model in random environments this paper uses regime switching diffusions to model the dynamics of the population sizes of n different species in an ecosystem subject to the random changes of the external environment it is demonstrated that the growth rates of the population sizes of the species are bounded above moreover certain long run average limits of the solution are examined from several angles a partial stochastic principle of competitive exclusion is also derived finally simple examples are used to demonstrate our findings

local continuity and asymptotic behaviour of degenerate parabolic systems abstract we study the local holder continuity and the asymptotic behaviour of solution u u 1 u k of the degenerate system u t i m u m 1 u i for m 1 and i 1 k which describes the population densities of k species whose diffusions are determined by their total population density u u 1 u k for the local holder continuity we adopt the intrinsic scaling and iteration arguments of degiorgi moser and dibenedetto under some regularity conditions we also prove that the population density of i th species with the population m i converges in c s to the function m i m b m x t as t where b m is the barenblatt profile of the standard porous medium equation with l 1 mass m m 1 m k as a consequence of asymptotic behaviour it is shown that each density becomes a concave function after a finite time

boundedness of the gradient for a doubly nonlinear parabolic equation we prove the local boundedness of the gradient for positive solutions to a doubly nonlinear parabolic equation in the case when the standard lebesgue measure has been replaced by a doubling measure which supports a weak poincare inequality

stabilisation of hybrid stochastic systems with levy noise by discrete time feedback control this paper is devoted to deal with the stabilisation problems of continuous time hybrid stochastic systems often described by stochastic differential equations with markovian switching with levy

self improving property of the fast diffusion equation abstract we show that the gradient of the m power of a solution to a singular parabolic equation of porous medium type also known as fast diffusion equation satisfies a reverse holder inequality in suitable intrinsic cylinders relying on an intrinsic calderon zygmund covering argument we are able to prove the local higher integrability of such a gradient for m n 2 n 2 1 our estimates are satisfied for a general class of growth assumptions on the non linearity in this way we extend the theory for m 1 see 16 in the list of references to the singular case in particular an intrinsic metric that depends on the solution itself is introduced for the singular regime

ergodic control of switching diffusions we study the ergodic control problem of switching diffusions representing a typical hybrid system that arises in numerous applications such as fault tolerant control systems flexible manufacturing systems etc under fairly general conditions we establish the existence of a stable nonrandomized markov policy which almost surely minimizes the pathwise long run average cost we then study the corresponding hamilton jacobi bellman hjb equation and establish the existence of a unique solution in a certain class using this we characterize the optimal policy as a minimizing selector of the hamiltonian associated with the hjb equations as an example we apply the results to a failure prone manufacturing system and obtain closed form solutions for the optimal policy

existence of weak solutions of doubly nonlinear parabolic equations abstract we deal with a cauchy dirichlet problem with homogeneous boundary conditions on the parabolic boundary of a space time cylinder for doubly nonlinear parabolic equations whose prototype is t u div u m 1 d u p 2 d u f with a non negative lebesgue function f on the right hand side where p 2 n n 2 and m 0 the central objective is to establish the existence of weak solutions under the optimal integrability assumption on the inhomogeneity f the constructed solution is obtained by a limit of approximations i e we use solutions of regularized cauchy dirichlet problems and pass to the limit to receive a solution for the original cauchy dirichlet problem

the conormal derivative problem for elliptic equations of variational type on considere des solutions du probleme qu 0 dans nu 0 sur ou est un domaine borne dans r n et q et n sont definis par qu x div a x u x du x b x u x du x x nu x a x u x du x x x x

riesz potentials and pointwise estimates of solutions to anisotropic porous medium equation abstract for an anisotropic porous medium equation with l 1 regular lower order terms we prove the local boundedness and continuity of weak solutions in terms of the linear riesz potential of the right hand side of equation

stabilization of linear systems by noise it is proved that the biggest lyapunov number lambda max of the system dot x a f t x where a is a fixed d times d matrix and f t is a zero mean strictly stationary matrix valued stochastic process satisfies 1 d trace a leqq lambda max on the other hand for each varepsilon 0 there is a process f t for which lambda max leqq 1 d trace a varepsilon in particular the system dot x ax can be stabilized by zero mean stationary parameter noise if and only if trace a 0 the stabilization can be accomplished by a one dimensional noise source the results carry over to the case where a is a stationary process they are also true for f t white noise

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

higher integrability for solutions to parabolic problems with irregular obstacles and nonstandard growth abstract the aim of this paper is to derive the self improving property of integrability for the spatial gradient of solutions to degenerate parabolic obstacle problem with irregular obstacles and p x t nonstandard growth more precisely we prove that the spatial gradient of the solution is integrable to a larger power than the natural one determined by the structural assumptions on the involved differential operator

regularization and stabilization of randomly switching dynamic systems this work concerns regularization or explosion suppression and stabilization of randomly switching dynamic systems by selecting suitable input functions starting with a randomly switching system modulated by a continuous time markov chain our aim is to find a feedback control input function so that the system becomes stable but before addressing the stabilization issue another problem needs to be resolved first because of the fast growth in the continuous state variable the solution of the system has a finite explosion time with probability one to ensure the existence of global solutions a feedback control a noise perturbation is added to make the system regular then another feedback control a second perturbing noise is added to ensure the stability of the resulting systems owing to the nonlinearity closed form solutions are often virtually impossible to obtain thus a discrete time approximation algorithm is constructed under simple conditions a suitably interpolated sequence of the

cauchy problem for fast diffusion equations with a gradient term we study the cauchy problem for fast diffusion equation with a gradient term of the form ut um duq where 0 0 and 0 0 however as q m the optimality of the stipulated assumptions is left open

the heat flow in nonlinear hodge theory we study the nonlinear hodge systemd 0 and 2 0 for an exterior form on a compact oriented riemannian manifold m where q is a given positive function the solutions are called harmonic forms they are the stationary points on cohomology classes of the functional e me 2 dm with e q q 2 the codifferential of a form is defined as 1 with 2 we evolve a given closed form 0 by the nonlinear heat flow system d for a time dependent exterior form x t on m this system is the differential of the normalized gradient flow u for e with 0 du under a technical assumption on the function 2 q q q we show that the nonlinear heat flow system d with initial condition 0 0 has a unique solution for all times which converges to a harmonic form in the cohomology class of 0 this yields a nonlinear hodge theorem that every cohomology class of m has a unique harmonic representative

holder estimates for parabolic p x t laplacian systems in this paper we establish the local holder continuity of the spatial gradient of weak solutions to the parabolic p x t laplacian system begin array lll partial t u rm div left a x t du p x t 2 du right 0 end array more precisely we prove that begin array lll du in c rm loc 0 alpha alpha 2 quad rm for some alpha in 0 1 end array provided p and a are holder continuous

hyperbolic phenomena in a strongly degenerate parabolic equation we consider the equation u t u u x x where 0 and where is a strictly increasing function with lim s we solve the associated cauchy problem for an increasing initial function and discuss to what extent the solution behaves qualitatively like solutions of the first order conservation law u t u x equations of this type arise for example in the theory of phase transitions where the corresponding free energy functional has a linear growth rate with respect to the gradient

stabilization and destabilization of nonlinear differential equations by noise this paper considers the stabilization and destabilization by a brownian noise perturbation that preserves the equilibrium of the ordinary differential equation x t f x t in an extension of earlier work we lift the restriction that f obeys a global linear bound and show that when f is locally lipschitz a function g can always be found so that the noise perturbation g x t db t either stabilizes an unstable equilibrium or destabilizes a stable equilibrium when the equilibrium of the deterministic equation is nonhyperbolic we show that a nonhyperbolic perturbation suffices to change the stability properties of the solution

local calderon zygmund estimates for parabolic minimizers abstract we establish local calderon zygmund estimates for parabolic minimizers u to certain variational inequalities the formal corresponding parabolic partial differential equation would be t u div f z u d u u f z u d u div f p 2 f h with p 2 n n 2 however since we do not impose any condition on the existence of the derivative u f the pde above may not have any meaning at all for proving the calderon zygmund estimates we thus remain solely on the level of parabolic minimizers precisely for q p we prove that d u l loc q if u is locally holder continuous and f l loc q and h l loc q for a q depending on q

travelling waves in a convection diffusion equation abstract we study existence and stability of travelling waves for nonlinear convection diffusion equations in the 1 d euclidean space the diffusion coefficient depends on the gradient in analogy with the p laplacian and may be degenerate unconditional stability is established with respect to initial data perturbations in l 1 r

holder zygmund estimates for degenerate parabolic systems abstract we consider energy solutions of the inhomogeneous parabolic p laplacian system t u div d u p 2 d u div g we show in the case p 2 that if the right hand side g is locally in l bmo then u is locally in l c 1 where c 1 is the 1 holder zygmund space this is the borderline case of the calderon zygmund theory we provide local quantitative estimates we also show that finer properties of g are conserved by du e g holder continuity moreover we prove a new decay for gradients of p caloric solutions for all 2 n n 2 p

the asymptotic properties of the suppressed functional differential system by brownian noise under regime switching abstractin the previous work by wu et al it was shown that the polynomial brownian noise may suppress the potential explosion of the nonlinear system without the linear growth condition or the one sided linear growth condition and the linear brownian noise may stabilise this suppressed system this paper is a continuation of our previous article and considers the asymptotic properties of the suppressed functional differential system under regime switching these asymptotic properties show that the suppressed functional differential system by polynomial brownian noise will grow with at most polynomial speed

regularity of solutions of nonlinear degenerate parabolic systems soit le systeme parabolique u i t div u p 2 u i 0 i m avec p 1 ou u i u i x t grad x et x r n ouvert pour toute solution de ce systeme avec max 1 2n n 2 p u est continu dans t