Some further results on the nonlocal p-Laplacian type problems

2020 ◽  
pp. 1-18
Author(s):  
Chao Zhang ◽  
Xia Zhang
Keyword(s):  
Weak Solutions ◽  
Integrable Function ◽  
Entropy Solutions ◽  
Existence Of Weak Solutions ◽  
Right Hand ◽  
The Right

We study the existence of entropy solutions by assuming the right-hand side function f to be an integrable function for some elliptic nonlocal p-Laplacian type problems. Moreover, the existence of weak solutions for the corresponding parabolic cases is also established. The main aim of this paper is to provide some positive answers for the two questions proposed by Chipot and de Oliveira (Math. Ann., 2019, 375, 283-306).

scholarly journals An extension of a Hardy-Littlewood-Pólya inequality

1978 ◽  
Vol 21 (1) ◽  
pp. 11-15 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Integrable Function ◽  
Right Hand ◽  
Usual Norm ◽  
Locally Integrable Function ◽  
Image Position ◽  
The Right

The Hardy-Littlewood-Pólya inequality in question can be written in the formHere and throughout, all functions are assumed to be locally integrable on ]0,∞[, 1≤p≤∞,p-1+(p′)-1=1 (with similar conventions for q,r,s), is the usual norm on Lp(0,∞), and if the right hand side is finite, then (1.1) is understood to mean thatdefines a locally integrable function Kf for which (1.1) holds.

Analysis of a system of nonlocal balance laws with weighted work in progress

2018 ◽  
Vol 15 (03) ◽  
pp. 375-406 ◽  
Author(s):  
Alexander Keimer ◽  
Günter Leugering ◽  
Tanmay Sarkar
Keyword(s):  
Weak Solutions ◽  
Nonlocal Term ◽  
Balance Laws ◽  
Source Terms ◽  
Existence Of A Solution ◽  
Existence Of Weak Solutions ◽  
Work In Progress ◽  
Proposed Model ◽  
The Right ◽  
Over Time

We consider a system of nonlocal balance laws, modeling a multi-commodity supply edge. The coupling of the different goods is realized via a “weighted work in progress” (WWIP), such that the velocity of every balance law also depends on the “load” of the other commodities. This WWIP represents nonlocal impact and due to it, the proposed model offers it to weight goods which are close to the “entrance” different from goods which leave the supply edge. It also allows the nonlocal term to have an impact on the solution only on a subset of the space domain, changing over time. Incorporating weights and source terms in the right-hand side, we demonstrate the existence of weak solutions for the resulting model and, under some restrictions on the weight and the inhomogeneity, we ensure the uniqueness of weak solutions. Furthermore, we consider source terms which depend pointwise on the solution and prove existence of a solution under suitable conditions.

scholarly journals Local subestimates of solutions to double-phase parabolic equations via nonlinear parabolic potentials

2019 ◽  
Vol 16 (1) ◽  
pp. 28-45
Author(s):  
Kateryna Buryachenko
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Growth Conditions ◽  
Local Boundedness ◽  
Nonstandard Growth ◽  
Right Hand ◽  
Double Phase ◽  
Nonlinear Parabolic ◽  
Nonstandard Growth Conditions ◽  
The Right

For parabolic equations with nonstandard growth conditions, we prove local boundedness of weak solutions in terms of nonlinear parabolic potentials of the right-hand side of the equation.

scholarly journals Weak solutions of complex Hessian equations on compact Hermitian manifolds

2016 ◽  
Vol 152 (11) ◽  
pp. 2221-2248 ◽  
Author(s):  
Sławomir Kołodziej ◽  
Ngoc Cuong Nguyen
Keyword(s):  
Weak Solutions ◽  
Stability Result ◽  
Existence Of Weak Solutions ◽  
Hermitian Manifolds ◽  
Positive Data ◽  
Right Hand ◽  
Hessian Equations

We prove the existence of weak solutions of complex $m$-Hessian equations on compact Hermitian manifolds for the non-negative right-hand side belonging to $L^{p}$, $p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has recently been solved by Székelyhidi and Zhang. We also give a stability result for such solutions.

scholarly journals Existence of entropy solutions for nonlinear elliptic equations in Musielak framework with L1 data

2018 ◽  
Vol 36 (1) ◽  
pp. 125 ◽  
Author(s):  
Elemine Vall Mohamed Saad Bouh ◽  
A. Ahmed ◽  
A. Touzani ◽  
A. Benkirane
Keyword(s):  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Nonlinear Elliptic Equations ◽  
Entropy Solutions ◽  
Strongly Nonlinear ◽  
Nonlinear Elliptic ◽  
Right Hand ◽  
L1 Data ◽  
The Right

We prove existence of solutions for strongly nonlinear elliptic equations of the form $$ \left\{\begin{array}{c} A(u)+g(x,u,\nabla u)=f+\mbox {div}(\phi(u))\quad \textrm{in }\Omega, \\ u\equiv0\quad \partial \Omega. \end{array} \right.$$ Where $A(u)=-\mbox {div}(a(x,u,\nabla u))$ be a Leray-Lions operator defined in $D(A)\subset W^{1}_{0}L_\varphi(\Omega) \rightarrow W^{-1}_{0}L_\psi(\Omega)$, the right hand side belongs in $ L^{1}(\Omega)$, and $\phi\in C^{0}(\mathbb{R},\mathbb{R}^N)$, without assuming the $\Delta_{2}$-condition on the Musielak function.

scholarly journals Existence of Solutions for Some Nonlinear Elliptic Anisotropic Unilateral Problems with Lower Order Terms

2018 ◽  
Vol 4 (2) ◽  
pp. 171-188 ◽  
Author(s):  
Youssef Akdim ◽  
Chakir Allalou ◽  
Abdelhafid Salmani
Keyword(s):  
Lower Order ◽  
Existence Of Solutions ◽  
Entropy Solutions ◽  
Unilateral Problem ◽  
Nonlinear Elliptic ◽  
Right Hand ◽  
Unilateral Problems ◽  
The Right ◽  
Lower Order Terms

AbstractIn this paper, we prove the existence of entropy solutions for anisotropic elliptic unilateral problem associated to the equations of the form$$ - \sum\limits_{i = 1}^N {{\partial _i}{a_i}(x,u,\nabla u) - } \sum\limits_{i = 1}^N {{\partial _i}{\phi _i}(u) = f,} $$where the right hand side f belongs to L1(Ω). The operator $- \sum\nolimits_{i = 1}^N {{\partial _i}{a_i}\left( {x,u,\nabla u} \right)} $ is a Leray-Lions anisotropic operator and ϕi ∈ C0(ℝ,ℝ).

Editorial Note

1946 ◽  
Vol 11 (1) ◽  
pp. 2-2
Keyword(s):  
Editorial Note ◽  
Speech Sounds ◽  
Left Hand ◽  
Hand Column ◽  
Right Hand ◽  
Infant Speech ◽  
The Right

In the article “Infant Speech Sounds and Intelligence” by Orvis C. Irwin and Han Piao Chen, in the December 1945 issue of the Journal, the paragraph which begins at the bottom of the left hand column on page 295 should have been placed immediately below the first paragraph at the top of the right hand column on page 296. To the authors we express our sincere apologies.

Eosinophilic vasculitis: an inhabitual and resistant manifestation of a vasculitis

VASA ◽  
2010 ◽  
Vol 39 (4) ◽  
pp. 344-348 ◽  
Author(s):  
Jandus ◽  
Bianda ◽  
Alerci ◽  
Gallino ◽  
Marone
Keyword(s):  
Skin Biopsy ◽  
Intravenous Iron ◽  
Small Vessel Vasculitis ◽  
High Dose ◽  
Raynaud Phenomenon ◽  
The Third ◽  
Left Elbow ◽  
Right Hand ◽  
The Right ◽  
Trunk Skin

A 55-year-old woman was referred because of diffuse pruritic erythematous lesions and an ischemic process of the third finger of her right hand. She was known to have anaemia secondary to hypermenorrhea. She presented six months before admission with a cutaneous infiltration on the left cubital cavity after a paravenous leakage of intravenous iron substitution. She then reported a progressive pruritic erythematous swelling of her left arm and lower extremities and trunk. Skin biopsy of a lesion on the right leg revealed a fibrillar, small-vessel vasculitis containing many eosinophils.Two months later she reported Raynaud symptoms in both hands, with a persistent violaceous coloration of the skin and cold sensation of her third digit of the right hand. A round 1.5 cm well-delimited swelling on the medial site of the left elbow was noted. The third digit of her right hand was cold and of violet colour. Eosinophilia (19 % of total leucocytes) was present. Doppler-duplex arterial examination of the upper extremities showed an occlusion of the cubital artery down to the palmar arcade on the right arm. Selective angiography of the right subclavian and brachial arteries showed diffuse alteration of the blood flow in the cubital artery and hand, with fine collateral circulation in the carpal region. Neither secondary causes of hypereosinophilia nor a myeloproliferative process was found. Considering the skin biopsy results and having excluded other causes of eosinophilia, we assumed the diagnosis of an eosinophilic vasculitis. Treatment with tacrolimus and high dose steroids was started, the latter tapered within 12 months and then stopped, but a dramatic flare-up of the vasculitis with Raynaud phenomenon occurred. A new immunosupressive approach with steroids and methotrexate was then introduced. This case of aggressive eosinophilic vasculitis is difficult to classify into the usual forms of vasculitis and constitutes a therapeutic challenge given the resistance to current immunosuppressive regimens.

Detection of Sarcoidosis Based on Subcutaneous Nodules in the Dorsum of the Right Hand

2014 ◽  
Vol 76 (1) ◽  
pp. 14-17
Author(s):  
Yoshiyuki KUWAE ◽  
Kunitaka HARUNA ◽  
Yasushi SUGA
Keyword(s):  
Subcutaneous Nodules ◽  
Right Hand ◽  
The Right
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