相关论文: Global solutions to special Lagrangian equations

We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must…

In this paper, we solve the Dirichlet problem for Lagrangian phase equation with critical and supercritical phase. A crucial ingredient is the interior $C^2$ estimate. Our result is sharp in the sense that there exist singular solutions in…

We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for…

Solutions to special Lagrangian equations near infinity, with supercritical phases or with semiconvexity on solutions, are known to be asymptotic to quadratic polynomials for dimension $n\ge 3$, with an extra logarithmic term for $n=2$. Via…

In this paper, we establish the existence and uniqueness theorem of entire solutions to the Lagrangian mean curvature equations with prescribed asymptotic behavior at infinity. The phase functions are assumed to be supercritical and…

We establish quadratic asymptotics for solutions to special Lagrangian equations with supercritical phases in exterior domains. The method is based on an exterior Liouville type result for general fully nonlinear elliptic equations toward…

We derive a priori interior Hessian and gradient estimates for special Lagrangian equation of phase at least a critical value in dimension three.

We establish a priori interior curvature estimates for the special Lagrangian curvature equations in both the critical phase and convex case. Additionally, we prove a priori interior gradient estimates for any constant phases.

We prove longtime existence and estimates for solutions to a fully nonlinear Lagrangian parabolic equation with locally $C^{1,1}$ initial data $u_0$ satisfying either (1) $-(1+\eta) I_n\leq D^2u_0 \leq (1+\eta)I_n$ for some positive…

We prove optimality conditions for generalized quantum variational problems with a Lagrangian depending on the free end-points. Problems of calculus of variations of this type cannot be solved using the classical theory.

We obtain an essentially optimal estimate for the moment of order 32/3 of the exponential sum having argument $\alpha x^3+\beta x^2$. Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine…

In this paper, we deal with two logarithmic fourth order differential equations: the extended one-dimensional DLSS equation and its multi-dimensional analog. We show the global existence of solution in critical spaces, its convergence to…

We prove the existence of infinite energy global solutions of the cubic wave equation in dimension greater than 3. The data is a typical element on the support of suitable probability measures.

We prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems of the calculus of variations on time scales. Here the Lagrangian depends on the independent variable, an unknown function and…

Some global existence criteria for quaternionic Riccati equations are established. Two of them are used to prove a completely non conjugation theorem for solutions of linear systems of ordinary differential equations.

In this paper we mainly investigate the periodic $\mu$-Camassa-Holm equation. We show the existence of global conservative solutions to the Cauchy problem of the periodic $\mu$-Camassa-Holm equation. The result is obtained by introducing a…

Using an rotation of Yuan, we observe that the gradient graph of any semiconvex function is a Liouville manifold, that is, does not admit bounded harmonic functions. As a corollary, we find that any entire solution of the fourth order…

We study 1D NLS with non-gauge invariant quadratic nonlinearity on the torus. The Cauchy problem admits trivial global solutions which are constant with respect to space. The non-existence of global solutions also has been studied only by…

In this paper, we establish interior Hessian and gradient estimates for the two-dimensional Lagrangian mean curvature equation when the phase changes signs, provided the gradient of the phase vanishes along its zero set. At the critical…

We classify global Lipschitz solutions to two-phase free boundary problems governed by concave fully nonlinear equations, as either two-plane solutions or solutions to a one-phase problem.