Related papers: Gradient variational problems in $\mathbb{R}^2$

We develop the calculus of variations on time scales for a functional that is the composition of a certain scalar function with the delta and nabla integrals of a vector valued field. Euler-Lagrange equations, transversality conditions, and…

Motivated by tug-of-war games and asymptotic analysis of certain variational problems, we consider a gradient constraint problem involving the infinity Laplace operator. We prove that this problem always has a solution that is unique if a…

This paper extends recent results on the exponential performance analysis of gradient based cooperative control dynamics using the framework of exponential integral quadratic constraints ($\alpha-$IQCs). A cooperative source-seeking problem…

From the recent developing of nonlocal gradients with finite horizon $\delta>0$ based on general kernels, we introduce a new nonlocal $p$-Laplacian and study the eigenvalue problem associated with it. Furthermore, by virtue of…

We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete…

This article establishes an interior gradient higher integrability result for weak solutions to parabolic multi-phase problems. The prototype equation for the parabolic multi-phase problem of $p$-Laplace type is given by \[ u_t -…

Conformal geodesics are solutions to a system of third order of equations, which makes a Lagrangian formulation problematic. We show how enlarging the class of allowed variations leads to a variational formulation for this system with a…

In this paper we study integro-differential equations like the anisotropic fractional Laplacian. As in [Silvestre, Indiana University Mathematics Journal 55, 2006], we adapt the De Giorgi technique to achieve the $C^{\gamma}$-regularity for…

We prove boundary higher integrability for the (spatial) gradient of \emph{very weak} solutions of quasilinear parabolic equations of the form $$ \left\{ \begin{array}{ll} u_t - div \mathcal{A}(x,t,\nabla u) = 0 &\quad \text{on} \ \Omega…

Some solutions for one class of nonlinear fourth-order partial differential equations \[u_{tt} = ({\kappa u + \gamma u^2})_{xx} + \nu uu_{xxxx} + \mu u_{xxtt} + \alpha u_x u_{xxx} + \beta u_{xx}^2 \] where $\alpha ,\;\beta ,\;\gamma ,\;\mu…

Causal variational principles, which are the analytic core of the physical theory of causal fermion systems, are found to have an underlying Hamiltonian structure, giving a formulation of the dynamics in terms of physical fields in…

We develop physically admissible lattice models in the harmonic approximation which define by Hamilton's variational principle fractional Laplacian matrices of the forms of power law matrix functions on the n -dimensional periodic and…

Any constructive continuous function must have a gradually varied approximation in compact space. However, the refinement of domain for $\sigma-$-net might be very small. Keeping the original discretization (square or triangulation), can we…

Using properties of harmonic functions in multidimensional space, we transform the Hartree-Fock eigenvalue problem into a more tractable eigenvalue problem in which the Laplacian is eliminated. This new formulation may facilitate the…

We examine several characteristics of conformal maps that resemble the variance of a Gaussian: asymptotic variance, the constant in Makarov's law of iterated logarithm and the second derivative of the integral means spectrum at the origin.…

In this paper, we investigate whether Variational Principles can be associated with the Helmholtz equation subject to impedance (absorbing) boundary conditions. This model has been extensively studied in the literature from both…

We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the…

The theory of quadrature domains for harmonic functions and the Hele-Shaw problem of the fluid dynamics are related subjects of the complex variables and mathematical physics. We present results generalizing the above subjects for elliptic…

In a recent letter ({\it{EPL}}, {\bf{104}} (2013) 60003; see also {\it {arXiv:1309.5645}}), Plastino and Rocca suggest that the divergences inherent to the formulation of nonextensive statistical mechanics can be eliminated {\it {via}} the…

We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and H\"{o}lder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable…