A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR, UNIFORMLY ELLIPTIC EQUATIONS

We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear,uniformly elliptic equations under Dirichlet boundary conditions. When ...     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator-prey models

This paper is concerned with two-species spatial homogeneous and inhomogeneous predator-prey models with Beddington-DeAngelis functional response. For the spatial homogeneous model, the asymptotic behavior of the interior equilibrium and the existence of Hopf bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered. Furthermore, the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are investigated. For the model with no-flux boundary conditions, Turing instability of the interior equilibrium solution is studied. In particular, Turing instability region regarding the parameters is established. Finally, to verify our theoretical results, some numerical simulations are also included.     

SOLUTION WITH SHOCK-BOUNDARY LAYER AND SHOCK-INTERIOR LAYER TO A CLASS OF NONLINEAR PROBLEMS

In this paper,the shock behaviors of solution to a class of nonlinear singularly perturbed problems are considered.Under some appropriate conditions,the outer and interior solutions to the original problem are constructed.Using the special limit and matching theory,the expressions of solutions with the shock behavior near the boundary and some interior points are given and the domain for boundary values is obtained.     

Nonlinear singular sturm-liouville problems and an application to transonic flow through a nozzle

We consider a class of singular Sturm-Liouville problems with a nonlinear convection and a strongly coupling source. Our investigation is motivated by, and then applied to, the study of transonic gas flow through a nozzle. We are interested in such solution properties as the exact number of solutions, the location and shape of boundary and interior layers, and nonlinear stability and instability of solutions when regarded as stationary solutions of the corresponding convective reaction-diffusion equations. Novel elements in our theory include a priori estimate for qualitative behavior of general solutions, a new class of boundary layers for expansion waves, and a local uniqueness analysis for transonic solutions with interior and boundary layers.     

On the Vekua Pair in Spheroidal Geometry and its Role in Solving Boundary Value Problems

The Vekua pair forms a transformation between the kernel of the Laplace's and the kernel of the Helmholtz's operator. In fact, it provides an interior solution of the Helmholtz's equation once an interior harmonic function is given, and conversely, given an interior solution of the Helmhotz's equation an interior harmonic function is constructed. Consequently, it seems that the Vekua connection offers the perfect ground to obtain solutions of boundary value problems connected with Helmholtz operator. Vekua expressed his transformation in spherical coordinates. Nevertheless, when a change of coordinates is applied, the transformation assumes a much more complicated form, but it still remains a very useful technique for dealing with solutions of the equations of Laplace and Helmholtz. Here we extend the Vekua theory to a new integral transformation pair concerning solutions of the aforementioned operators in exterior domains. In addition, the form of the Vekua transformation is analyzed in spheroidal coordinates and its implication to boundary value problems is investigated.     

On the Interaction Between Dissipation and Supply Agents in Some Nonlinear Evolution Problems

The paper deals with typical examples of semilinear evolution systems possessing solutions which behave at the interior of the considered domain unlike they do at its boundary. Sharp conditions arc obtained which guarantee interior boundedness of a solution component despite the growth of corresponding data as well as ensure strict interior positivity of a solution component despite its decay at the boundary.     

Interior regularity of solutions to a complex Monge-Ampère equation

We give interior estimates for first derivatives of solutions to a type of complex Monge-Ampère equations in convex domains. We also show global estimates for first derivatives of solutions in arbitrary domains. These global estimates are then used to show interior regularity of solutions to the complex Monge-Ampère equations in hyperconvex domains having a bounded exhaustion function which is globally Lipschitz. Finally we give examples of domains which have such an exhaustion function and domains which do not. The author was partially supported by the Royal Swedish Academy of Sciences, Gustaf Sigurd Magnuson’s fund.     

Symmetric Euler and Navier-Stokes shocks in stationary barotropic flow on a bounded domain

We construct stationary solutions to the barotropic, compressible Euler and Navier-Stokes equations in several space dimensions with spherical or cylindrical symmetry. For given Dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the Euler equations in the exterior domain. On the other hand, stationary smooth solutions in the interior domain necessarily become sonic and cannot be continued beyond a critical inner radius. We then use these solutions to construct entropy-satisfying shocks for the Euler equations in the region between two concentric spheres or cylinders. Next we construct smooth Navier-Stokes solutions converging to the previously constructed Euler shocks in the small viscosity limit. In the process we introduce a new technique for constructing smooth solutions, which exhibit a fast transition in the interior, to a class of two-point boundary problems.     

Existence and Asymptotic Behavior of the Wave Equation with Dynamic Boundary Conditions

The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time.     

Multi-valued solutions to Hessian equations

In this paper, we use the Perron method to prove the existence of bounded multi-valued viscosity solutions to Hessian equations and interior Lipschitz continuity of the multi-valued solutions.     

Interior regularity of weak solutions to the perturbed Navier-Stokes equations

In this paper we establish interior regularity for weak solutions and partial regularity for suitable weak solutions of the perturbed Navier-Stokes system, which can be regarded as generalizations of the results in L. Caffarelli, R. Kohn, L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure. Appl. Math. 35 (1982), 771–831, and S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscr. Math. 69 (1990), 237–254.     

Solutions with many mixed positive and negative interior spikes for a semilinear Neumann problem

We study the existence of sign-changing multiple interior spike solutions for the following Neumann problem $$\varepsilon^2\Delta v-v+f(v) = 0 \,\, {\rm in} \,\, \Omega, \quad \frac{\partial v}{\partial \nu} = 0 \,\, {\rm on} \,\, \partial \Omega,$$ where ?? is a smooth bounded domain of ${\mathbb {R}^N}$ , ?? is a small positive parameter, f is a superlinear, subcritical and odd nonlinearity. No symmetry on ?? is assumed. To our knowledge, only positive interior peak solutions have been obtained for this problem and it remains a question whether or not multiple interior peak solutions with mixed positive and negative peaks exist. In this paper we assume that ?? is a two-dimensional strictly convex domain and, provided that k is sufficiently large, we construct a (k?+?1)-peak solutions with k positive interior peaks aligned on a closed curve near ??? and 1 negative interior peak located in a more centered part of ??.     

A priori estimate for convex solutions to special Lagrangian equations and its application

We derive a priori interior Hessian estimates for special Lagrangian equations when the potential is convex. When the phase is very large, we show that continuous viscosity solutions are smooth in the interior of the domain. © 2008 Wiley Periodicals, Inc.     

On a class of interior point algorithms

A family of interior point algorithms for solving linear programs is examined. Under the assumption on the nondegeneracy of the problem, a theoretical justification of these algorithms is given. The sets of the algorithms converging to relatively interior optimal solutions and having linear or superlinear convergence rate are identified.     

On the wave equation with semilinear porous acoustic boundary conditions

The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function.     

一类具有非单调内层性态的半线性边值问题

本文运用内层校正方法和微分不等式理论研究了一类半线性边值问题.在一定的条件下,我们获得了两类非单调内层性态:尖层性态或非单调过渡层性态的解的一致有效复合展开式.     

Regularity for Almost Convex Viscosity Solutions of the Sigma-2 Equation

We establish interior regularity for almost convex viscosity solutions of the sigma-2 equation.     

Nontrivial solutions for Monge-Ampère type operators in convex domains

In this paper we prove existence and interior regularity of convex solutions of the Dirichlet problem for a class of Monge-Ampère type equations with right hand side vanishing on the boundary, by means of the fixed point index theory for Bakelman's weak solutions. The regularity in the interior follows from recent results of Caffarelli. Partially supported by M.U.R.S.T., Italy Partially supported by G.N.A.F.A. of C.N.R., Italy. This work was accomplished during a Humbold research stay at Karlsruhe and Heidelberg Universities.     

SEPARABLE CONDITIONS AND SCALARIZATION OF BENSON PROPER EFFICIENCY

Under the assumption that the ordering cone has a nonempty interior and is separable （or the feasible set has a nonempty interior and is separable）, we give scalarization theorems on Benson proper effciency. Applying the results to vector optimization problems with nearly cone-subconvexlike set-valued maps, we obtain scalarization theorems and Lagrange multiplier theorems for Benson proper effcient solutions.