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Register a new userSome recent results on the Dirichlet problem for (p,q)-Laplace equations
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A short account of recent existence and multiplicity theorems on the Dirichlet problem for an elliptic equation with $(p,q)$-Laplacian in a bounded domain is performed. Both eigenvalue problems and different types of perturbation terms are briefly discussed. Special attention is paid to possibly coercive, resonant, subcritical, critical, or asymmetric right-hand sides.
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This thesis is concerned with the asymptotic behavior of solutions of stochastic $p$-Laplace equations driven by non-autonomous forcing on $mathbb{R}^n$. Two cases are studied, with additive and multiplicative noise respectively. Estimates on the tails of solutions are used to overcome the non-compactness of Sobolev embeddings on unbounded domains, and prove asymptotic compactness of solution operators in $L^2(mathbb{R}^n)$. Using this result we prove the existence and uniqueness of random attractors in each case. Additionally, we show the upper semicontinuity of the attractor for the multiplicative noise case as the intensity of the noise approaches zero.
The aim of this paper is to develop the regularity theory for a weak solution to a class of quasilinear nonhomogeneous elliptic equations, whose prototype is the following mixed Dirichlet $p$-Laplace equation of type begin{align*} begin{cases} mathrm{div}(| abla u|^{p-2} abla u) &= f+ mathrm{div}(|mathbf{F}|^{p-2}mathbf{F}) qquad text{in} Omega, hspace{1.2cm} u &= g hspace{3.1cm} text{on} partial Omega, end{cases} end{align*} in Lorentz space, with given data $mathbf{F} in L^p(Omega;mathbb{R}^n)$, $f in L^{frac{p}{p-1}}(Omega)$, $g in W^{1,p}(Omega)$ for $p>1$ and $Omega subset mathbb{R}^n$ ($n ge 2$) satisfying a Reifenberg flat domain condition or a $p$-capacity uniform thickness condition, which are considered in several recent papers. To better specify our result, the proofs of regularity estimates involve fractional maximal operators and valid for a more general class of quasilinear nonhomogeneous elliptic equations with mixed data. This paper not only deals with the Lorentz estimates for a class of more general problems with mixed data but also improves the good-$lambda$ approach technique proposed in our preceding works~cite{MPT2018,PNCCM,PNJDE,PNCRM}, to achieve the global Lorentz regularity estimates for gradient of weak solutions in terms of fractional maximal operators.
We introduce Fundamental solutions of Barenblatt type for the equation $u_t=sum_{i=1}^N bigg( |u_{x_i}|^{p_i-2}u_{x_i} bigg)_{x_i}$, $p_i >2 quad forall i=1,..,N$, on $Sigma_T=mathbb{R}^N times[0,T]$, and we prove their importance for the regularity properties of the solutions.
We prove a boundary Harnack principle in Lipschitz domains with small constant for fully nonlinear and $p$-Laplace type equations with a right hand side, as well as for the Laplace equation on nontangentially accessible domains under extra conditions. The approach is completely new and gives a systematic approach for proving similar results for a variety of equations and geometries.
In this paper, the author studies quaternionic Monge-Amp`ere equations and obtains the existence and uniqueness of the solutions to the Dirichlet problem for such equations without any restriction on domains. Our paper not only answers to the open problem proposed by Semyon Alesker in [3], but also extends relevant results in [7] to the quaternionic vector space.
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