scholarly journals Critical perturbations of elliptic operators by lower order terms

2021 ◽  
Author(s):  
◽  
Jose Luis Luna-Garcia
Keyword(s):  
Fundamental Solution ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
The Other ◽  
Layer Potentials ◽  
Lebesgue Spaces ◽  
Uniqueness Of Solutions ◽  
Linear Elliptic Operator ◽  
Lower Order Terms ◽  
Associated Solution

In this work we study issues of existence and uniqueness of solutions of certain boundary value problems for elliptic equations in the upper half-space. More specifically we treat the Dirichlet, Neumann, and Regularity problems for the general second order, linear, elliptic operator under a smallness assumption on the coefficients in certain critical Lebesgue spaces. Our results are perturbative in nature, asserting that if a certain operator L[subscript 0] has good properties (as far as boundedness and invertibility of certain associated solution operators), then the same is true for L[subscript 1], whenever the coefficients of these two operators are close in certain L[subscript p] spaces. Our approach is through the theory of layer potentials, though the lack of good estimates for solutions of L [equals] 0 force us to use a more abstract construction of these objects, as opposed to the more classical definition through the fundamental solution. On the other hand, these more general objects suggest a wider range of applications for these techniques. The results contained in this thesis were obtained in collaboration with Simon Bortz, Steve Hofmann, Svitlana Mayboroda, and Bruno Poggi. The resulting publications can be found in [BHL+a] and [BHL+b].

EXISTENCE OF SOLUTIONS FOR A CONTINUOUS MULTIGRAIN MODEL FOR POLYMERIZATION

2000 ◽  
Vol 10 (08) ◽  
pp. 1263-1276
Author(s):  
DANIELE ANDREUCCI ◽  
ANTONIO FASANO ◽  
RICCARDO RICCI
Keyword(s):  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
The Other ◽  
Uniqueness Of Solutions ◽  
Small Times ◽  
Catalyst Pellets ◽  
Multigrain Model ◽  
Diffusion Problems

We prove the existence and uniqueness of solutions, for small times, for a mathematical scheme modeling the Ziegler–Natta process of polymerization. The model consists, essentially, of two diffusion problems at two different space scales, one relative to the microscopical catalyst pellets, the other to the macroscopical aggregate of those pellets. The coupling between the two scales is of nonstandard nature.

Existence results for the Dirichlet problem of some degenerate nonlinear elliptic equations

2014 ◽  
Vol 20 (2) ◽  
Author(s):  
Albo Carlos Cavalheiro
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Nonlinear Elliptic Equations ◽  
Existence Results ◽  
Uniqueness Of Solutions ◽  
Nonlinear Elliptic

AbstractIn this paper we are interested in the existence and uniqueness of solutions for the Dirichlet problem associated to the degenerate nonlinear elliptic equations

scholarly journals Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients

2018 ◽  
Vol 24 (1) ◽  
pp. 105-127
Author(s):  
Fabio Punzo ◽  
Enrico Valdinoci
Keyword(s):  
Asymptotic Behavior ◽  
Parabolic Equations ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Time Dependent ◽  
Uniqueness Of Solutions ◽  
Unbounded Coefficients ◽  
Conditions At Infinity

We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed point-wise conditions at infinity (in space), which can be time-dependent. Moreover, we study the asymptotic behavior of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.

Existence and uniqueness of solutions of an A-harmonic elliptic equation

2019 ◽  
Vol 56 (1) ◽  
pp. 13-21
Author(s):  
Mouna Kratou
Keyword(s):  
Weak Solution ◽  
Elliptic Equation ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions

Abstract This paper deals with the existence and uniqueness of weak solution of a problem which involves a class of A-harmonic elliptic equations of nonhomogeneous type. Under appropriate assumptions on the function f, our main results are obtained by using Browder Theorem.

scholarly journals Existence and uniqueness of solutions for a semilinear elliptic system

2005 ◽  
Vol 2005 (10) ◽  
pp. 1507-1523 ◽  
Author(s):  
Robert Dalmasso
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Biharmonic Equation ◽  
Nonlinear Elliptic Equations ◽  
Uniqueness Of Solutions ◽  
Nonlinear Elliptic ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic System ◽  
Homogeneous Dirichlet Problem

We consider the existence, the nonexistence, and the uniqueness of solutions of some special systems of nonlinear elliptic equations with boundary conditions. In a particular case, the system reduces to the homogeneous Dirichlet problem for the biharmonic equationΔ2u=|u|pin a ball withp>0.

scholarly journals Comparison Principle for Elliptic Equations with Mixed Singular Nonlinearities

2021 ◽  
Author(s):  
Riccardo Durastanti ◽  
Francescantonio Oliva
Keyword(s):  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Elliptic Boundary ◽  
Laplacian Operator ◽  
Bounded Subset ◽  
Elliptic Boundary Value Problem ◽  
Lebesgue Spaces ◽  
Elliptic Boundary Value ◽  
Singular Nonlinearities ◽  
Open Bounded Subset

AbstractWe deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by $$ \left \{\begin {array}{ll} \displaystyle -{\Delta }_{p} u= \frac {f}{u^{\gamma }} + g u^{q} & \text { in } {\Omega }, \\ u = 0 & \text {on } \partial {\Omega }, \end {array}\right . $$ − Δ p u = f u γ + g u q in Ω , u = 0 on ∂ Ω , where Ω is an open bounded subset of $\mathbb {R}^{N}$ ℝ N where Ω is an open bounded subset of $\mathbb {R}^{N}$ ℝ N , Δpu := ÷(|∇u|p− 2∇u) is the usual p-Laplacian operator, γ ≥ 0 and 0 ≤ q ≤ p − 1; f and g are nonnegative functions belonging to suitable Lebesgue spaces.

scholarly journals Existence and uniqueness of solutions for a class of degenerate nonlinear elliptic equations

2016 ◽  
Vol 70 (2) ◽  
pp. 9
Author(s):  
Albo Carlos Cavalheiro
Keyword(s):  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Weighted Sobolev Spaces ◽  
Nonlinear Elliptic Equations ◽  
Uniqueness Of Solutions ◽  
Nonlinear Elliptic

In this work we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations \begin{align} {\Delta}(v(x)\, {\vert{\Delta}u\vert}^{p-2}{\Delta}u) &-\sum_{j=1}^n D_j{\bigl[}{\omega}_1(x) \mathcal{A}_j(x, u, {\nabla}u){\bigr]}+ b(x,u,{\nabla}u)\, {\omega}_2(x)\\ & = f_0(x) - \sum_{j=1}^nD_jf_j(x), \ \ {\rm in } \ \ {\Omega} \end{align} in the setting of the weighted Sobolev spaces.

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