Search

The objective of this paper is two-fold. The first objective is to complete the former work of Bensoussan and Frehse [2] . One big limitation of this paper was the fact that they are systems of PDE. on a bounded domain. One can expect solutions to be bounded, since one looks for smooth solutions. This is a very important property for the development of the method. It is true also that solutions which exist in a bounded domain may fail to exist on R n , because of the lack of bounds. We give conditions so that the results of [2] can be extended to R n . The second objective is to consider the BSDE (Backward stochastic differential equations) version of the system of PDE. This is the objective of a more recent work of Xing and Zitkovic [8] . They consider systems of BSDE with quadratic growth, which is a well-known open problem in the BSDE literature. Since the BSDE are Markovian, the problem is equivalent to the analytic one. However, because of this motivation the analytic problem is in R n and not on a bounded domain. Xing and Zitkovic developed a probabilistic approach. The connection between the analytic problem and the BSDE is not apparent. Our objective is to show that the analytic approach can be completely translated into a probabilistic one. Nevertheless probabilistic concepts are also useful, after their conversion into the analytic framework. This is in particular true for the uniqueness result.

The authors deal with nonlinear elliptic and parabolic systems that are the Bellman like systems associated to stochastic differential games with mean field dependent dynamics. The key novelty of the paper is that they allow heavily mean field dependent dynamics. This in particular leads to a system of PDE’s with critical growth, for which it is rare to have an existence and/or regularity result. In the paper, they introduce a structural assumptions that cover many cases in stochastic differential games with mean field dependent dynamics for which they are able to establish the existence of a weak solution. In addition, the authors present here a completely new method for obtaining the maximum/minimum principles for systems with critical growths, which is a starting point for further existence and also qualitative analysis.

We consider stochastic optimal control problems with an additional term representing the variance of the control functions. The latter one may serve as a risk control. We present and treat the prob...

Since its introduction by P.L. Lions in his lectures and seminars at the College de France, see Lions [6], and also the very helpful notes of Cardialaguet (2013) on Lions’ lectures, the Master Equation has attracted a lot of interest, and various points of view have been expressed, see for example Carmona and Delarue (2014), Bensoussan et al. (2015), Buckdahn et al. [3]. There are several ways to introduce this type of equation; and in those mentioned works, they involve an argument which is a probability measure, while P.L. Lions has recently proposed the alternative idea of working on the Hilbert space of square integrable random variables. Hence writing the equation is an issue; while another issue is its origin. In this article, we discuss all these various aspects. An important reference is the seminar at College de France delivered by P.L. Lions on November 14, 2014.

We consider the necessary conditions for Nash-points of Vlasov-McKean functionals $$\mathcal {J}^{i}[\mathbf{v}]=\int _{Q}mf^{i}(\cdot ,m,\mathbf{v})\,dx\,dt$$Ji[v]=?Qmfi(·,m,v)dxdt ($$i=1,...,N$$i=1,...,N). The corresponding payoffs $$f^{i}$$fi depend on the controls $$\mathbf{v}$$v and, in addition, on the field variable $$m=m(\mathbf{v})$$m=m(v). The necessary conditions lead to a coupled forward-backward system of nonlinear parabolic equations, motivated by stochastic differential games. The payoffs may have a critical nonlinearity of quadratic growth and any polynomial growth w.r.t. m is allowed as long as it can be dominated by the controls in a certain sense. We show existence and regularity of solutions to these mean-field-dependent Bellman systems by a purely analytical approach, no tools from stochastics are needed.

We study here quasi-linear parabolic equations with quadratic growth in \(\mathbb {R}^{n}\). These parabolic equations are at the core of the theory of PDE; see Friedman (Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, 1964) [6], Ladyzhenskaya et al. (Translations of Mathematical Monographs. AMS, 1968) [4] for details. However, for the applications to physics and mechanics, one deals mostly with boundary value problems. The boundary is often taken to be bounded and the solution is bounded. This brings an important simplification. On the other hand, stochastic control theory leads mostly to problems in \(\mathbb {R}^{n}\). Moreover, the functions are unbounded and the Hamiltonian may have quadratic growth. There may be conflicts which prevent solutions to exist. In stochastic control theory, a very important development deals with BSDE (Backward Stochastic Differential Equations). There is a huge interaction with parabolic PDE in \(\mathbb {R}^{n}\). This is why, although we do not deal with BSDE in this paper, we use many ideas from Briand and Hu (Probab Theory Relat Fields 141(3–4):543–567, 2008) [1], Da Lio and Ley (SIAM J Control Optim 45(1):74–106, 2006) [2], Karoui et al. (Backward stochastic differential equations and applications, Princeton BSDE Lecture Notes, 2009) [3], Kobylanski (Ann Probab 28(2):558–602, 2000) [5]. Our presentation provided here is slightly innovative.

We consider two most studied standard models in the theory of elasto-plasticity in arbitrary dimension $$d\ge 2$$ , namely, the Hencky model and the Prandtl–Reuss model subjected to the von Mises condition. There are many available results for these models—from the existence and the regularity theory up to the relatively sharp identification of the plastic strain in the natural function/measure space setting. In this paper we shall proceed further and improve some of known estimates in order to identify sharply the plastic strain. More specifically, we rigorously improve the integrability of the displacement and the velocity (which was known only under a nonnatural assumption that the Cauchy stress is bounded), show the BMO estimates for the stress and finally also the Morrey-like estimates for the plastic strain. In addition, we shall provide the whole theory up to the boundary. As an immediate consequence of such improved estimates, we provide a sharper identification of the plastic strain than that known up to date. In particular, in two dimensional setting, we show that the plastic strain can be point-wisely characterized in terms of the stresses everywhere although the stress is possibly discontinuous and thus the natural duality pairing in the space of measures could be violated.

We consider a general Bellman type system of parabolic partial differential equations with a special coupling in the zero order terms. We show the existence of solutions in Lp((0,T);W2,p(O))nW1,p((0,T)×O) by establishing suitable a priori bounds. The system is related to a certain non zero sum stochastic differential game with a maximum of two players. The players control a diffusion in order to minimise a certain cost functional. During the game it is possible that present players may die or a new player may appear. We assume that the death, respectively the birth time of a player is exponentially distributed with intensities that depend on the diffusion and the controls of the players who are alive.

We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) $p$-Laplace system and models for non-Newtonian fluids of powerlaw or Carreau type. By the use of special weak different quotients adapted to the variational structure we bound fractional derivatives of $u_t$ in time and space directions. Although the estimates presented here are valid under very general assumptions they are a novelty even for the parabolic $p$-Laplace equation.

We consider weak solutions of nonlinear elliptic systems in a $$W^{1,p}$$ -setting which arise as Euler–Lagrange equations of certain variational integrals with pollution term, and we also consider minimizers of a variational problem. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the independent and the dependent variables. We impose new structural conditions on the nonlinearities which yield $$\fancyscript{C}^{\alpha }$$ -regularity and $$\fancyscript{C}^{\alpha }$$ -estimates for the solutions. These structure conditions cover variational integrals like $$\int F(\nabla u)\,\mathrm{d}x $$ with potential $$F(\nabla u):=\tilde{F} (Q_1(\nabla u),\ldots , Q_N(\nabla u))$$ and positive definite quadratic forms $$Q_i$$ in $$\nabla u$$ defined as $$Q_i(\nabla u)=\sum \nolimits _{\alpha \beta } a_i^{\alpha \beta } \nabla u^\alpha \cdot \nabla u^\beta $$ . A simple example consists in $${\tilde{F}}(\xi _1,\xi _2):= |\xi _1|^{\frac{p}{2}} + |\xi _2|^{\frac{p}{2}}$$ or $$\tilde{F}(\xi _1,\xi _2):= |\xi _1|^{\frac{p}{4}}|\xi _2|^{\frac{p}{4}}.$$ Since the quadratic forms $$Q_i$$ need not to be linearly dependent, our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. As a by-product, we also prove a kind of Liouville theorem. As a new analytical tool, we use a weighted integral technique with singular weights in an $$L^p$$ -setting for the proof and establish a weighted hole-filling inequality in a setting where Green-function techniques are not available.

We consider a non zero sum stochastic differential game with a maximum $n$ players, where the players control a diffusion in order to minimise a certain cost functional. During the game it is possible that present players may die or new players may appear. The death, respectively the birth time of a player is exponentially distributed with intensities that depend on the diffusion and the controls of the players who are alive. We show how the game is related to a system of partial differential equations with a special coupling in the zero order terms. We provide an existence result for solutions in appropriate spaces that allow to construct Nash optimal feedback controls. The paper is related to a previous result in a similar setting for two players leading to a parabolic system of Bellman equations [4]. Here, we study the elliptic case (infinite horizon) and present the generalisation to more than two players.

Mean field theory has raised a lot of interest in the recent years (see in particular the results of Lasry-Lions in 2006 and 2007, of Gueant-Lasry-Lions in 2011, of Huang-Caines-Malham in 2007 and many others). There are a lot of applications. In general, the applications concern approximating an infinite number of players with common behavior by a representative agent. This agent has to solve a control problem perturbed by a field equation, representing in some way the behavior of the average infinite number of agents. This approach does not lead easily to the problems of Nash equilibrium for a finite number of players, perturbed by field equations, unless one considers averaging within different groups, which has not been done in the literature, and seems quite challenging. In this paper, the authors approach similar problems with a different motivation which makes sense for control and also for differential games. Thus the systems of nonlinear partial differential equations with mean field terms, which have not been addressed in the literature so far, are considered here.

The authors consider quasilinear parabolic systems $$ {\partial _t}u - div\,a\left( {\nabla u} \right) = 0. $$ in two space dimensions. The function a has p-growth behaviour, 1< p < ∞, and the ellipticity “constant” behaves like (1+|∇u|)p− 2. The author prove full regularity of the weak solution on interior subdomains, but globally in time. The key idea in the proof is a technique to obtain boundedness of the gradient based on logarithmic estimates.

We obtain everywhere 𝒞α-regularity for vector solutions to a class of nonlinear elliptic systems whose principal part is the Euler operator to a variational integral ∫ F ( u , ∇ u ) d x ${\int F(u,\nabla u)\, dx}$ with quadratic growth in ∇ u ${\nabla u}$ and which satisfies a generalized splitting condition that cover the case F ( u , ∇ u ) : = ∑ i Q i , $ {F(u,\nabla u):= \sum _i Q_i},\vspace*{-2.27621pt} $ where Q i : = ∑ α β A i α β ( u , ∇ u ) ∇ u α · ∇ u β ${Q_i := \sum _{\alpha \beta } A^{\alpha \beta }_i(u,\nabla u) \nabla u^{\alpha } \cdot \nabla u^{\beta }}$ , or the case F ( u , ∇ u ) : = ∏ i ( 1 + Q i ) θ i . $ {F(u,\nabla u) := \prod _i (1+Q_i)^{\theta _i}}.\vspace*{-2.13394pt} $ A crucial assumption is the one-sided condition F u ( u , η ) · u ≥ - K $ F_u (u,\eta ) \cdot u \ge -K\vspace*{-0.62596pt} $ and related generalizations. In the elliptic case we obtain existence of 𝒞α-solutions. If the leading operator is not necessarily elliptic but coercive, possible minima are everywhere Hölder continuous and the same holds also for Noether solutions, i.e., extremals which are also stationary with respect to inner variations. In particular if A α β ( u , ∇ u ) = A α β ( u ) ${A^{\alpha \beta }(u,\nabla u)=A^{\alpha \beta } (u)}$ , our result generalizes a result of Giaquinta and Giusti. The technique of our proof (using weighted norms and inhomogeneous hole-filling method) does not rely on L ∞ ${L^{\infty }}$ -a priori estimates for the solution.

A large class of stochastic differential games for several players is considered in this paper. The class includes Nash differential games as well as Stackelberg differential games. A mix is possible. The existence of feedback strategies under general conditions is proved. The limitations concern the functionals in which the state and the controls appear separately. This is also true for the state equations. The controls appear in a quadratic form for the payoff and linearly in the state equation. The most serious restriction is the dimension of the state equation, which cannot exceed 2. The reason comes from PDE (partial differential equations) techniques used in studying the system of Bellman equations obtained by Dynamic Programming arguments. In the authors’ previous work in 2002, there is not such a restriction, but there are serious restrictions on the structure of the Hamiltonians, which are violated in the applications dealt with in this article.

We deal with nonlinear elliptic and parabolic systems that are the Bellman systems associated to stochastic differential games as a main motivation. We establish the existence of weak solutions in any dimension for an arbitrary number of equations ("players"). The method is based on using a renormalized sub- and super-solution technique . The main novelty consists in the new structure conditions on the critical growth terms with allow us to show weak solvability for Bellman systems to certain classes of stochastic differential games.

We consider the Navier–Stokes equations for compressible isentropic flow in the steady three-dimensional case and show the existence of a weak solution for homogeneous Dirichlet (no-slip) boundary conditions under the assumption that the adiabatic exponent satisfies γ > 4 3 . In particular we cover with our existence result the cases of a monoatomic gas ( γ = 5 3 ) and of air ( γ = 7 5 ). To our knowledge it is the first result that really deals in 3-D with the existence of a weak solution in these physically relevant cases with arbitrary large external data and these boundary conditions. As an essential tool we demonstrate and use a weighted estimate respective an estimate in a Morrey-space for the pressure and resulting from this an L ∞ -estimate for the inverse Laplacian of the pressure.

We consider the loading of an elastic perfectly plastic body governed by the Prandtl-Reuss law. It is shown that the stress velocities of the body have fractional derivatives of order - δ up to the boundary in the direction of the loading parameter, and of order ⅓ - δ in the interior of the body in direction of the space variables.

Problems of elastic plastic deformation with kinematic or isotropic hardening and von Mises flowrule are considered. It is shown that the velocities of the stresses, strains and hardening parameters satisfy an improved Lp-property that is σ, ξ, ∇u ∈ L∞(0, T; L2+2δ(Ω)) with some δ > 0. For dimension n = 2 this implies continuity of u in spatial direction, furthermore it can be used as tool to prove boundary differentialbility σ, ξ ∈ L∞(0, T; L1+e) and σ, ξ ∈ L∞(0, T; 1/2+δ ′,2), where 1/2 + δ ′ is the order of fractional Nichol'skii differentiability (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We consider weak solutions to nonlinear elliptic systems in a W1,p-setting which arise as Euler equations to certain variational problems. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the dependent and independent variables. We impose new structure conditions on the coefficients which yield everywhere \({\mathcal{C}^{\alpha}}\)-regularity and global \({\mathcal{C}^{\alpha}}\)-estimates for the solutions. These structure conditions cover variational integrals like \({\int F(\nabla u)\; dx}\) with potential \({F(\nabla u):=\tilde F (Q_1(\nabla u),\ldots, Q_N(\nabla u))}\) and positively definite quadratic forms in \({\nabla u}\) defined as \({Q_i(\nabla u)=\sum_{\alpha \beta} a_i^{\alpha \beta} \nabla u^\alpha \cdot \nabla u^\beta}\). A simple example consists in \({\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{2}} + |\xi_2|^{\frac{p}{2}}}\) or \({\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{4}}|\xi_2|^{\frac{p}{4}}}\). Since the Qi need not to be linearly dependent our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. The proof uses a new weighted norm technique with singular weights in an Lp-setting.

We consider nonlinear elliptic Bellman systems which arise in the theory of stochastic differential games. The right-hand sides of the equations (which are called Hamiltonians) may have quadratic growth with respect to the gradient of the unknowns. Under certain assumptions on the Hamiltonians, that are satisfied for many types of stochastic games, we establish the existence of a regular solution. This partially generalizes recent works of Bensoussan, Frehse and Vogelgesang from two-dimensional setting onto arbitrary dimension. The main novelty of the paper consists of introducing a new (semi-continuity) method for obtaining the continuity of the solution and the corresponding estimates.

We consider parabolic systems u t − div(a(∇u)) = f in two space dimensions where the elliptic part is derived from a potential and is coercive, but not monotone. With natural assumptions on the data we obtain the existence of a long-time Holder continuous solution in the sense of Young measures.

We consider unsteady flows of inhomogeneous, incompressible, shear-thickening and heat-conducting fluids where the viscosity depends on the density, the temperature and the shear rate, and the heat conductivity depends on the temperature and the density. For any values of initial total mass and initial total energy we establish the long-time existence of weak solution to internal flows inside an arbitrary bounded domain with Lipschitz boundary.

We consider the Navier–Stokes equations for compressible isothermal flow in the steady two-dimensional case and show the existence of a weak solution for homogeneous Dirichlet boundary conditions.

We consider a class of non-linear partial differential systems like -div$(a(x)\nabla u_{\nu}) +\lambda u_{\nu}=H_{\nu}(x, Du) \, $ with applications for the solution of stochastic differential games with $N$ players, where $N$ is an arbitrary but positive number. The Hamiltonian $H$ of the non-linear system satisfies a quadratic growth condition in $D u$ and contains interactions between the players in the form of non-compact coupling terms $\nabla u_{i} \cdot\nabla u_j$. A $L^{\infty}\cap H^1$-estimate and regularity results are shown, mainly in two-dimensional space. The coupling arises from cyclic non-market interaction of the control variables.

We investigate regularity properties of the stress tensor near the boundary for models of elasto-plasticity in arbitrary dimension. Focusing on special geometries, namely on balls and infinite strips, we obtain L 2-estimates for the tangential derivatives of the stress tensor near the boundary. We indicate why these estimates may fail for more general domains. In addition, we establish L 2-estimates for (the trace of) the stress tensor on the boundary.

We consider the Navier–Stokes equations for compressible isothermal flow in the steady two dimensional case and show the existence of a weak solution in the case of mixed boundary conditions. The proof is based on an analysis of the stream function, exploiting the convective term.

We consider parabolic systems $$u_{t} - {\rm div} \left( a(t, x, u, \nabla u)\right) + a_{0}(t, x, u, \nabla u) = 0$$ in two space dimensions with initial and Dirichlet boundary conditions. The elliptic part including a0 is derived from a potential with quadratic growth in ∇u and is coercive and monotone. The term a0 may grow quadratically in ∇u and satisfies a sign condition a0 · u ≥ −K. We prove the existence of a regular long time solution verifying a regularity criterion of Arkhipova. No smallness is assumed on the data.

We consider in this article a class of systems of second order partial differential equations with non-linearity in the first order derivative and zero order term which can be super-quadratic. These problems are motivated by differential geometry and stochastic differential games. Up to now, in the case of systems, only quadratic growth had been considered.

We consider the regularity of weak solutions to evolution variational inequalities arising from the flow theory of plasticity with isotropic and kinematic hardening. The (linear) elasticity tensor is allowed to have discontinuities. We derive a Morrey condition for the stress velocities and the strains (not the strain velocity!) up to the boundary. In the case of two space dimensions we conclude the Holder continuity of the displacements.

We consider mixtures of compressible viscous fluids consisting of two miscible species. In contrast to the theory of non-homogeneous incompressible fluids where one has only one velocity field, here we have two densities and two velocity fields assigned to each species of the fluid. We obtain global classical solutions for quasi-stationary Stokes-like system with interaction term.

We present an irregular weak solution $${u : \mathbb {R}^n \to \mathbb {C}}$$ of a uniformly elliptic scalar equation in divergence form with measurable coefficients. The solution has a square integrable gradient. Such examples have been known for dimension n ≥ 5 only.

We show the existence of weak solutions to the system describing the motion of incompressible, non-homogeneous generalized Newtonian fluids if the extra stress tensor S(ρ, D) possesses p-structure with $${p \ge \frac {3d+2}{d+2}}$$ and variable viscosity. The limiting process in the equation of motion is justified by a variational argument, which is new in this context.

We consider quasilinear diagonal elliptic systems in bounded domains subject to Dirichlet, Neumann or mixed boundary conditions. The leading elliptic operator is assumed to have only measurable coefficients, and the nonlinearities (Hamiltonians) are allowed to be of quadratic (critical) growth in the gradient variable of the unknown. These systems appear in many applications, in particular in differential geometry and stochastic differential game theory. We impose on the Hamiltonians structural conditions developed between 1972–2002 and also a new condition (sum coerciveness) introduced in recent years (in the context of the pay off functional in stochastic game theory). We establish existence, Holder continuity, Liouville properties, W2,q estimates, etc. for solutions, via a unified approach through the blow-up method. The main novelty of the paper is the introduction of a completely new technique, which in particular leads to smoothness of the solution also for dimensions d≥3.

Solutions to nonlinear partial differential equations of fourth order are studied. Boundary regularity is proved for solutions that satisfy mixed boundary conditions. Various geometric situations including so called triple points are considered. Regularity is measured in Sobolev-Slobodeckii spaces and the results are sharp in this scale. The approach is based on the use of a first order difference quotient method.

We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities ϱi of the fluids and their velocity fields u(i) are prescribed at infinity: ϱ i|∞ = ϱi∞ > 0, u(i)|∞ = 0. Neglecting the convective terms, we have proved earlier that weak solutions to such a reduced system exist. Here we establish a uniqueness type result: in the absence of the external forces and interaction terms, there is only one such solution, namely ϱi ≡ ϱi∞, u(i) ≡ 0, i = 1, 2.

We consider the Navier–Stokes equations for compressible isentropic flow in the steady three-dimensional case. The pressure ργ and the kinetic energy Open image in new window are estimated uniformly in Lq with Open image in new windowρ being the density. This is an improvement of known estimates in the case Open image in new window

We consider a simplified model that describes steady flows of a miscible mixture of fluids. The corresponding system of equations is studied in the whole space $\R^3$. The densities $\rho_i$ of the species and their velocity fields $\ui$ are prescribed at infinity: $\rho_i|_{\infty} = \rho_{i \infty} > 0\,,\;\; \ui|_{\infty} = 0$. We prove the existence of weak solutions to the system under consideration.

We deal with a system of partial differential equations describing a steady motion of an incompressible fluid with shear-dependent viscosity and present a new global existence result for $ p>\frac{2d}{d+2} $. Here p is the coercivity parameter of the nonlinear elliptic operator related to the stress tensor and d is the dimension of the space. Lipschitz test functions, a subtle splitting of the level sets of the maximal functions for the velocity gradients, and a decomposition of the pressure are incorporated to obtain almost everywhere convergence of the velocity gradients.

We consider in this article diagonal parabolic systems arising in the context of stochastic dierential games. We address the issue of nding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic dierential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear Hamiltonian seems to be the adequate one to achieve the regularity property. A key step in the theory is to prove the existence of Holder solution. Mathematics Subject Classication. 35XX, 49XX.

In this article we consider local solutions for stochastic Navier Stokes equations, based on the approach of Von Wahl, for the deterministic case. We present several approaches of the concept, depending on the smoothness available. When smoothness is available, we can in someway reduce the stochastic equation to a deterministic one with a random parameter. In the general case, we mimic the concept of local solution for stochastic differential equations.

We study the question of integrability for the gradient of the stress near the boundary for models of elasto-perfect plasticity and their approximations.

The nonlinear elliptic equation is investigated. It is supposed that u fulfils a mixed boundary value condition and that Ω ⊂ IRn (n ≥ 3) has a piecewise smooth boundary. Ws,2 — regularity (s < 3/2) of u and Lp — properties of the first and the second derivatives of u are proven.