Search results for "Laplacian"
showing 5 items of 135 documents
Location of solutions for quasi-linear elliptic equations with general gradient dependence
2017
Existence and location of solutions to a Dirichlet problem driven by $(p,q)$-Laplacian and containing a (convection) term fully depending on the solution and its gradient are established through the method of subsolution-supersolution. Here we substantially improve the growth condition used in preceding works. The abstract theorem is applied to get a new result for existence of positive solutions with a priori estimates.
On elliptic equations involving the 1-Laplacian operator
2018
El objetivo de esta tesis doctoral es dar a conocer los resultados obtenidos sobre existencia, unicidad y regularidad de las soluciones de diferentes ecuaciones elípticas regidas por el operador 1-laplaciano. El primer capítulo está dedicado al estudio de la ecuación - div (Du/|Du|) + g(u) |Du| = f(x) en un subconjunto abierto y acotado U de R^N con frontera Lipschitz, con la condición de Dirichlet u=0 en la frontera, tomando una función f positiva y siendo g una función real, continua y positiva. Por un lado, obtenemos soluciones no acotadas cuando el dato f pertenece al espacio de Marcinkiewicz L^{N,\infty}(U), por lo que debemos introducir la definición apropiada para este tipo de soluci…
Equivalence of viscosity and weak solutions for a $p$-parabolic equation
2019
AbstractWe study the relationship of viscosity and weak solutions to the equation $$\begin{aligned} \smash {\partial _{t}u-\varDelta _{p}u=f(Du)}, \end{aligned}$$ ∂ t u - Δ p u = f ( D u ) , where $$p>1$$ p > 1 and $$f\in C({\mathbb {R}}^{N})$$ f ∈ C ( R N ) satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $$p\ge 2$$ p ≥ 2 .
Hölder regularity for the gradient of the inhomogeneous parabolic normalized p-Laplacian
2018
In this paper, we study an evolution equation involving the normalized [Formula: see text]-Laplacian and a bounded continuous source term. The normalized [Formula: see text]-Laplacian is in non-divergence form and arises for example from stochastic tug-of-war games with noise. We prove local [Formula: see text] regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration.
Remarks on regularity for p-Laplacian type equations in non-divergence form
2018
We study a singular or degenerate equation in non-divergence form modeled by the $p$-Laplacian, $$-|Du|^\gamma\left(\Delta u+(p-2)\Delta_\infty^N u\right)=f\ \ \ \ \text{in}\ \ \ \Omega.$$ We investigate local $C^{1,\alpha}$ regularity of viscosity solutions in the full range $\gamma>-1$ and $p>1$, and provide local $W^{2,2}$ estimates in the restricted cases where $p$ is close to 2 and $\gamma$ is close to 0.