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The paper concerns regularities of solutions of elliptic equations in non-divergence form with directional homogenization. The main interest is to investigate the differences of regularities of datums that are needed and of solutions that are obtained between directions with homogenization and without homogenization. We establish uniform interior L p ( 1 p ≤ ∞ ) estimates for D 2 u e and uniform interior C γ ( 0 ≤ γ 1 ) estimates for D D x ′ u e , where x ′ are the non-homogenization directions. Our results are optimal.

In this paper we prove Krylov's boundary gradient type estimates (and regularity) for solutions to fully nonlinear differential inequalities with unbounded coefficients and quadratic growth on the ...

In this paper, we prove the hot spots conjecture for long rotationally symmetric domains in R n by the continuity method. More precisely, we show that the odd Neumann eigenfunction in x n associated with lowest nonzero eigenvalue is a Morse function on the boundary, which has exactly two critical points and is monotone in the direction from its minimum point to its maximum point. As a consequence, we prove that the Jerison and Nadirashvili's Conjecture 8.3 holds true for rotationally symmetric domains and are also able to obtain a sharp lower bound for the Neumann eigenvalue.

Boundary differentiability of solution of elliptic equation is shown while the boundary satisfies the exterior Reifenberg $$C^{1,Dini}$$ condition and is punctually $$C^{1}$$ at boundary point.

We show the global regularity of the nonhomogeneous parabolic p -Laplace equations on convex domains. Starting with the local up-to-boundary Lipschitz regularity for the homogeneous equation, we derive the global a priori estimates via Vitali covering coupling with p-harmonic approximation. Moreover, the exponents in the a priori estimates come from the intrinsic scaling.

In this paper, we investigate the robust utility maximization problems under both preferences of extremely ambiguity loving and ambiguity aversion. By a fundamental martingale characterization technique on nonlinear expectations, optimal investment strategies are explicitly solved in a general non-Markovian framework via backward stochastic differential equations. Different with previous works in the literature assuming the convexity of the set of prior probability measures , our analysis are independent of the cardinality of . Our results show that extremely ambiguity-loving (resp. -aversion) investors will adopt the extremely aggressive (resp. conservative) investment strategy.

For fully nonlinear uniformly parabolic equations, the first derivatives regularity of viscosity solutions at lateral boundary is studied under new Dini type conditions for the boundary, which is called Reifenberg Dini conditions and is weaker than usual Dini conditions.

In this paper we study a model which describes the pattern formation of vegetation spots and strips in a semi-arid or arid landscape. The mathematical model consists of a nonlinear cross-diffusion system with evaporation and absorption sources. Global existence and uniqueness in classical sense for the system are established. Some asymptotic behaviors of the solution are derived for a linearized system. It is shown that the solution of the linearized system is asymptotically stable near the steady-state solution. Moreover, we show that the Turing phenomenon occurs for the linearized cross-diffusion system and the cross-diffusion destabilizes the ecosystem.

In this paper we establish the existence and uniqueness of weak solutions for the initial-boundary value problem of a two-dimensional low-curvature equation, which is related to image analysis.

The purpose of this paper is to investigate positive solutions of integral equations involving Bessel potential. Exploiting the moving plane method in integral form, we give the radial symmetry of both the domain and solutions of our integral equations in exterior domains and in annular domains respectively.

This paper investigates oscillatory traveling wave solutions to an attractive chemotaxis system. The convective part of this system changes its type when crossing a parabola in the phase space. The oscillatory nature of the traveling wave comes from the fact that one far-field state is in the elliptic region and another in the hyperbolic region. Such traveling wave solutions are shown to be linearly unstable. Detailed construction of some traveling wave solutions is presented.

In this paper we obtain the following global a priori Calderon–Zygmund estimates in a convex domain Ω | f | p 1 + | f | p 2 ∈ L γ ( Ω ) ⇒ | ∇ u | p 1 + | ∇ u | p 2 ∈ L γ ( Ω ) for any γ ≥ 1 of weak solutions for a class of non-uniformly nonlinear elliptic equations with vanishing Dirichlet data ∑ i = 1 2 div ( ( A i ∇ u ⋅ ∇ u ) p i − 2 2 A i ∇ u ) = ∑ i = 1 2 div ( | f | p i − 2 f ) , where 1 p 1 p 2 ∞ .

We introduce asymptotically elliptic equations and establish the global W 2 , p estimates for viscosity solutions to both asymptotically linear elliptic equations and asymptotically fully nonlinear elliptic equations in bounded domains. Our approach for the proof is based on an appropriate transformation which converts a given asymptotically elliptic equation to a suitable uniformly elliptic equation.

In this article, we introduce a nonlinear expectation, called g*-expectation, based on g-expectation and consider the optimal utility under g*-expectation in the market with a risk-free bond and d risky stocks in finite trading interval [0, T]. We construct a stochastic family by taking advantage of the comparison theorem of backward stochastic differential equations and the g*-martingale. We generalize the results of Hu et al. (Annals of Applied Probability 28(2):1691–1712, 2005), and obtain the explicit forms of the optimal trading strategies both for exp -utility and the power utility, when g(t, z) = βt|z|2 + γtz.

This paper investigates the positive solutions for second order linear elliptic equation in unbounded cylinder with zero boundary condition. We prove there exist two special positive solutions with exponential growth at one end while exponential decay at the other, and all the positive solutions are linear combinations of these two.

In this paper, we consider the positive viscosity solutions for certain fully nonlinear uniformly elliptic equations in unbounded cylinder with zero boundary condition. After establishing an Aleksandrov-Bakelman-Pucci maximum principle, we classify all positive solutions as three categories in unbounded cylinder. Two special solution spaces (exponential growth at one end and exponential decay at the another) are one dimensional, independently, while solutions in the third solution space can be controlled by the solutions in the other two special solution spaces under some conditions, respectively.

We show global regularity for p -Laplacian type elliptic equations on convex domains with W 1 , q ( q ≥ p > 1 ) boundary data. The maximal function, Vitali covering lemma and approximation (or compactness method) are the main techniques in the proof. To derive the approximation lemma, we need W 1 , ∞ estimates for the corresponding homogeneous equations, where the convexity of the domain plays an important role.

We consider fourth order elliptic systems in divergence form with greatly discontinuous coefficients in a nonsmooth domain. Our domain is composed of a finite number of disjoint subdomains with ( δ , R ) -Reifenberg flat boundaries, and the coefficients, which are allowed to have great discontinuity across the boundaries of subdomains, have small BMO semi-norms on each subdomain. Our main result is deriving the global W 2 , p ( 1 p ∞ ) regularity, with the estimates independent of the distance between the subdomains.

We study a general model of nonvariational elliptic equations of \(p\)-Laplacian type in quasiconvex domains, which are locally approximated by convex domains. We prove that both the gradient and the associated nonhomogeneous term belong to the same \(L^q\) space for every \(q \in [p, \infty )\). As far as the domain is concerned, our regularity assumption on the boundary is weaker than any other one reported in this direction. In addition, we extend our result in Lebesgue spaces to Orlicz spaces.

In this work we prove the interior L q estimates for the weak solution of a class of p -Laplace type Baouendi–Grushin equations by the compactness method and its intrinsic metric associated with the geometry of the Baouendi–Grushin vector fields.

In this paper, we establish a new proof of L p estimates of the Stokes equations, which enables us to avoid the use of potential theory and thus simplifies the traditional proof. The modified Vitali covering lemma and the maximal function technique are the main analytical tools.

In this paper, we will analyze the blow-up behavior of solution sequences satisfying a conformal invariant equation defined on a compact 2-dimensional surface ( M , g ) with boundary. We will provide some accurate point-wise estimates for the profile of these sequences.

In this paper we consider the global gradient estimates for weak solutions of $p(x)$-Laplacian type equation with small BMO coefficients in a $\delta$-Reifenberg flat domain. The modified Vitali covering lemma, good $\lambda$-inequalities, the maximal function technique and the appropriate localization method are the main analytical tools. The global Calderon--Zygmund theory for such equations is obtained. Moreover, we generalize the regularity estimates in the Lebesgue spaces to the Orlicz spaces.

In this paper, we prove a Hausdorff measure estimate for the free boundaries of subsolutions of fully nonlinear and quasilinear equations of the type \({F(D^2u,x)\geqq f(x)}\) and \({{\rm div}\,A(x,\nabla u)\geqq \mu}\) where \({f \in L^{q}, q >N}\) and μ is a signed Radon measure with some appropriate growth condition. Gradient estimates for nonnegative harmonic functions with bounded normal derivatives along the boundary obtained by Caffarelli and Salsa (Geometric Approach to Free Boundary Problems, 2005) are extended to the context of inhomogeneous problems involving fully nonlinear and p-Laplace equations. As an application, Lipschitz regularity is obtained for one phase solutions of inhomogeneous nonlinear free boundary problems.

We establish global W 1, p(·)-estimates for second order elliptic equations in divergence form under the natural assumption that p(·) is log-Holder continuous. To this end, we assume that the coefficients are measurable in one variable and have small BMO semi-norms in the other variables and the boundary of the domain is Reifenberg flat. Our work is an optimal and natural extension of W 1,p -regularity for such equations with merely measurable coefficients beyond Lipschitz domains.

In this paper we obtain the global regularity estimates of the weak solutions in Sobolev spaces and Orlicz spaces for higher order elliptic and parabolic equations of divergence form with small BMO coefficients in the whole space. We only focus on the parabolic case while the corresponding result in the elliptic case can be obtained as a corollary.

The lateral boundary differentiability is shown for solutions of parabolic differential equations in nondivergence form under the assumptions that the parabolic boundary satisfies the exterior Dini condition and is punctually C 1 differentiable one-sided in t -direction. The classical barrier technique, the maximum principle, the interior Harnack inequality and an iteration procedure are the main analytical tools.

We investigate an overdetermined problem for the torsion equation Δ u = − 1 with two kinds of free boundary conditions in an unbounded domain and present a symmetry result in the case of 2D. The proof is based on the maximum principle. In the proof, we use the sliding method and some convex arguments.

A new proof for boundary Lipschitz estimate is shown for solutions of elliptic equations in nondivergence form. Under additional condition, the boundary differentiability is obtained.

In this paper, we focus on pricing European options in a double exponential jump diffusion model with stochastic volatility and stochastic interest rate. Firstly, using fast Fourier transform (FFT) technique, we obtain numerical solutions for option prices. Then, we analyze several effects on option prices under the proposed model, including correlation between stock returns and volatility, stochastic interest rate. Simulations show that FFT is fast and efficient, stock returns are negatively correlated with volatility and the effect of stochastic interest rate over longer time horizons is significant.

The lateral boundary differentiability with respect to x is shown for solutions of parabolic differential equations in nondivergence form on cylindrical convex domains.

In this paper we study the regularity theory of the solutions of a class of degenerate elliptic equations in divergence form. By introducing a proper distance and applying the compactness method we establish the Holder type estimates for the weak solutions.

In this paper, we study solutions of one phase inhomogeneous singular perturbation problems of the type: \( F(D^2u,x)=\beta _{\varepsilon }(u) + f_{\varepsilon }(x) \) and \( \Delta _{p}u=\beta _{\varepsilon }(u) + f_{\varepsilon }(x)\), where \(\beta _{\varepsilon }\) approaches Dirac \(\delta _{0}\) as \(\varepsilon \rightarrow 0\) and \(f_{\varepsilon }\) has a uniform control in \(L^{q}, q>N.\) Uniform local Lipschitz regularity is obtained for these solutions. The existence theory for variational (minimizers) and non variational (least supersolutions) solutions for these problems is developed. Uniform linear growth rate with respect to the distance from the \(\varepsilon -\)level surfaces are established for these variational and nonvaritional solutions. Finally, letting \(\varepsilon \rightarrow 0\) basic properties such as local Lipschitz regularity and non-degeneracy property are proven for the limit and a Hausdorff measure estimate for its free boundary is obtained.

This paper deals with the chemotaxis system $$ \left\{ \begin{array}{ll} u_t ={D} u_{xx}-\chi [u(\ln v)_x]_x, & x\in (0, 1), \ t>0,\\ v_t =\varepsilon v_{xx} +uv-\mu v, & x\in (0, 1), \ t>0, \end{array} \right. $$ under Neumann boundary condition, where $\chi0$, $\varepsilon>0$ and $\mu>0$ are constants. It is shown that for any sufficiently smooth initial data $(u_0, v_0)$ fulfilling $u_0\ge 0$, $u_0 \not\equiv 0$ and $v_0>0$, the system possesses a unique global smooth solution that enjoys exponential convergence properties in $L^\infty(\Omega)$ as time goes to infinity, which depend on the sign of $\mu-\bar{u}_0$, where $\bar{u}_0 :=\int_0^1 u_0 dx$. Moreover, we prove that the constant pair $(\mu, (\frac{\mu}{\lambda})^{\frac{D}{\chi}})$ (where $\lambda>0$ is an arbitrary constant) is the only positive stationary solution. The biological implications of our results will be given in the paper.

In this paper we obtain the interior \(C^{1,\alpha }\) regularity of weak solutions for a class of non-uniformly nonlinear elliptic equations $$\begin{aligned} \text {div} ~\! \left( a_1\left( \left| \nabla u \right| \right) \nabla u + a_2\left( \left| \nabla u \right| \right) \nabla u \right) =0, \end{aligned}$$ including the following special model $$\begin{aligned} \text {div} ~\! \left( \left| \nabla u \right| ^{p-2} \nabla u + \left| \nabla u \right| ^{q-2} \nabla u \right) =0\quad \ \text{ for } \text{ any } \ p, q>1. \end{aligned}$$ These equations come from variational problems whose model energy functional is given by $$\begin{aligned} \mathcal {P}(u, \Omega )=: \int _{\Omega } B^1\left( \left| \nabla u \right| \right) + B^{2}\left( \left| \nabla u \right| \right) dx, \end{aligned}$$ where $$\begin{aligned} B^k(t)=\int _0^t \tau a_k(\tau )~d\tau \quad \text{ for } \quad t\ge 0 \quad \text{ and }\quad k=1,2. \end{aligned}$$ We remark that $$\begin{aligned} B^k(t)= |t|^{\alpha _k} \log \big ( 1+|t|\big ) \quad \text{ for } ~~ ~~\alpha _k>1~~~ \text{ and }~~k=1,2 \end{aligned}$$ satisfy the given conditions in this work.

In this paper, we prove the estimates of modulus of continuity up to the boundary for the first order derivative of viscosity solutions for fully nonlinear uniformly elliptic equations under Dini boundary data with the domain in the same class. As a corollary we derive C 1 , α boundary regularity.

In this paper we obtain a Caccioppoli type estimate for the model of the linearized Mullins-Sekerka equations by a new technique, then we use this estimate to derive it's Schauder type estimates by polynomial approximation method.

The present paper studies the Dirichlet problem for elliptic equations degenerate on part of the boundary of a domain and the degeneracy is of the Keldysh type. By introducing a proper metric that is related to the operator we establish the global Holder estimates when some well-posed boundary conditions are satisfied. The main methods are the construction of some barrier functions and the interpolation of the estimates of uniformly elliptic operators.

In this paper, we investigate positive solutions of the following integral equation system { u ( x ) = ∫ R n | x − y | α − n u a ( y ) v b ( y ) d y , v ( x ) = ∫ R n | x − y | β − n | u c ( y ) v d ( y ) d y where a , b , c , d , α , β are constants. The symmetry and monotonicity of its solutions are proved by the method of moving planes, the non-existence and the exact form are given by the method of moving spheres.

In this paper we study a 2-dimensional parabolic variational inequality with financial background. We define a suitable weak formula and obtain existence and uniqueness of the problem. Moreover we analyze the behaviors of the free boundary surface.

We study a very general model of nonvariational elliptic equations of p-Laplacian type. We prove the W 1,q -estimates for each q > p regarding such a nonlinear elliptic equation in divergence form under the assumption that for each point and for each sufficiently small scale the nonlinearity is suitably close to a vector of the gradient in BMO space. In the same spirit our results can be easily extended to Orlicz spaces as well.

We treat fourth order parabolic systems in divergence form with bounded measurable coefficients. We establish Hessian estimates for such parabolic systems by proving that the Lp-regularity of the inhomogeneous terms exactly reflects in the regularity of the Hessian of the solutions for every \({p \in (1, \infty)}\) . The assumptions are that the coefficients are allowed to be merely measurable in one of spatial variables, but averaged in the other spatial variables and time variable.

We consider the following free boundary problem in an unbounded domain Ω in two dimensions: Δ p u = 0 in Ω, u = 0 , ∂ u ∂ n = g 0 on J 0 , u = 1 , ∂ u ∂ n = g 1 on J 1 , where ∂ Ω = J 0 ∪ J 1 . We prove that if 0 u 1 in Ω, J i is the graph of a function in C loc 1 , α ( R ) and g i is a constant for each i = 0 , 1 , then the free boundary ∂Ω must be two parallel straight lines and the solution u must be a linear function. The proof is based on maximum principle.

The main purpose of this paper is to investigate positive solutions of the following system of integral equations with weight Riesz potential on Ω: $$\left\{\begin{array}{ll} u(x)=\int\limits_{\Omega}{|x|^{-\alpha}|x-y|^{-\mu}|y|^{-\beta}}{u^a(y)v^b(y)}dy,\\ v(x)=\int\limits_{\Omega}{|x|^{-\gamma}|x-y|^{-\nu}|y|^{-\kappa}}{u^c(y)v^d(y)}dy,\\ \end{array} \right.$$ where a, b, c, d, α, β, γ, κ, μ, ν are constants. With the method of moving planes, we establish the symmetry of both the solution and the bounded C1 domain Ω if u and v are constants on ∂Ω. Moreover, we also give a symmetry result for systems of integral equations with weight Riesz potential on exterior domains.

In this paper, we investigate the symmetry of domains and solutions of integral equation systems on bounded domains. Under some natural integrability conditions, we prove that the domains are balls, all positive solutions of systems are radially symmetric and monotone decreasing with respect to the radius.

In this paper, we derive the global regularity of weak solutions to divergence form elliptic equations in the Orlicz space W 0 1 L ϕ with ϕ ∈ ∇ 2 ∩ Δ 2 . The domain considered here is quasiconvex, which is a generalization of the Reifenberg flat domain.