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## Concavity maximum principle for viscosity solutions of singular equations

### Nonlinear Differential Equations and Applications NoDEA (2010-10-01) 17: 601-618 , October 01, 2010

We prove a concavity maximum principle for the viscosity solutions of certain fully nonlinear and singular elliptic and parabolic partial differential equations. Our results parallel and extend those obtained by Korevaar and Kennington for classical solutions of quasilinear equations. Applications are given in the case of the singular infinity Laplace operator.

## Non-local heat flows and gradient estimates on closed manifolds

### Journal of Evolution Equations (2009-08-26) 9: 787-807 , August 26, 2009

In this paper, we study two kinds of *L*^{2} norm preserved non-local heat flows on closed manifolds. We first study the global existence, stability, and asymptotic behavior of such non-local heat flows. Next we give the gradient estimates of positive solutions to these heat flows.

## Stability of radial symmetry for a Monge-Ampère overdetermined problem

### Annali di Matematica Pura ed Applicata (2008-09-17) 188: 445-453 , September 17, 2008

Recently the symmetry of solutions to overdetermined problems has been established for the class of Hessian operators, including the Monge-Ampère operator. In this paper we prove that the radial symmetry of the domain and of the solution to an overdetermined Dirichlet problem for the Monge-Ampère equation is stable under suitable perturbations of the data.

## Radial entire solutions for supercritical biharmonic equations

### Mathematische Annalen (2006-04-01) 334: 905-936 , April 01, 2006

We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find finite time blow up solutions. Finally, we study the convergence at infinity of smooth solutions towards the explicitly known singular solution. It turns out that the convergence is different in space dimensions *n* ≤ 12 and *n* ≥ 13.

## Convex hull property and maximum principle for finite element minimisers of general convex functionals

### Numerische Mathematik (2013-08-01) 124: 685-700 , August 01, 2013

The convex hull property is the natural generalization of maximum principles from scalar to vector valued functions. Maximum principles for finite element approximations are often crucial for the preservation of qualitative properties of the respective physical model. In this work we develop a convex hull property for $$\mathbb{P }_1$$ conforming finite elements on simplicial non-obtuse meshes. The proof does not resort to linear structures of partial differential equations but directly addresses properties of the minimiser of a convex energy functional. Therefore, the result holds for very general nonlinear partial differential equations including e.g. the $$p$$ -Laplacian and the mean curvature problem. In the case of scalar equations the introduce techniques can be used to prove standard discrete maximum principles for nonlinear problems. We conclude by proving a strong discrete convex hull property on strictly acute triangulations.

## Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels

### Manuscripta Mathematica (2012-11-01) 139: 291-319 , November 01, 2012

In this paper, we consider fully nonlinear integro-differential equations with possibly nonsymmetric kernels. We are able to find different versions of Alexandroff–Backelman–Pucci estimate corresponding to the full class
$${\mathcal {S}^{\mathfrak {L}_0}}$$
of uniformly elliptic nonlinear equations with 1 < *σ* < 2 (subcritical case) and to their subclass
$${\mathcal {S}_{\eta}^{\mathfrak {L}_0}}$$
with 0 < *σ* ≤ 1. We show that
$${\mathcal {S}_{\eta}^{\mathfrak {L}_0}}$$
still includes a large number of nonlinear operators as well as linear operators. And we show a Harnack inequality, Hölder regularity, and *C*^{1,α}-regularity of the solutions by obtaining decay estimates of their level sets in each cases.

## Prescribing curvature problem of Bakry-Émery Ricci tensor

### Science China Mathematics (2013-09-01) 56: 1935-1944 , September 01, 2013

We consider the problem of deforming a metric in its conformal class on a closed manifold, such that the *k*-curvature defined by the Bakry-Émery Ricci tensor is a constant. We show its solvability on the manifold, provided that the initial Bakry-Émery Ricci tensor belongs to a negative cone. Moveover, the Monge-Amp`ere type equation with respect to the Bakry-Émery Ricci tensor is also considered.

## Nonautonomous and non periodic Schrödinger equation with indefinite linear part

### Journal of Fixed Point Theory and Applications (2017-03-01) 19: 17-36 , March 01, 2017

The existence of solution of the nonlinear Schrödinger equation
$$\begin{aligned} \begin{array}{lc} -\Delta u + V(x) u = f(x,u),&\end{array} \end{aligned}$$
is stablished in
$$\mathbb {R}^N$$
, where *V* changes sign and *f* is an asymptotically linear function at infinity, with *V* and *f* non periodic in *x*. Spectral theory, a classical linking theorem and interaction between translated solutions of the problem at infinity are employed.

## The G Class of Functions and Its Applications

### Acta Mathematica Sinica (2000-07-01) 16: 455-468 , July 01, 2000

###
*Abstract*

In this paper, we introduce the concept of the *G* class of functions of the parabolic class, and show the Hölder continuity of the *G* class of functions. The introduction of this concept contributes to the proof of the regularity and existence of the solution for the first boundary problem of parabolic equation in divergence form.

## Concentration-compactness principle for an inequality by D. Adams

### Calculus of Variations and Partial Differential Equations (2014-09-01) 51: 195-215 , September 01, 2014

This paper brings a generalization of the Lions concentration–compactness principle to the Sobolev space $$W_0^{m,p}(\Omega )$$ when $$mp=n$$ and $$\Omega \subset \mathbb {R}^n \, (n \ge 2)$$ is a smooth domain with finite $$n$$ -measure. Moreover, our result sharpens an inequality by D. Adams improving its best exponent. The proof is established using the decreasing rearrangement and comparison principle due to Talenti combined with the maximum principle. It is also shown how this concentration–compactness principle yields the existence of critical points of mountain-pass type for an energy functional associated with a class of polyharmonic problems.