scholarly journals Global Schauder Estimates for the $$\mathbf {p}$$-Laplace System

2021 ◽  
Author(s):  
D. Breit ◽  
A. Cianchi ◽  
L. Diening ◽  
S. Schwarzacher
Keyword(s):  
Global Regularity ◽  
Regularity Theory ◽  
Divergence Form ◽  
Dirichlet Problems ◽  
Schauder Estimates ◽  
First Order ◽  
Right Hand ◽  
Regularity Results ◽  
The Right ◽  
Linear Setting

AbstractAn optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the p-Laplace equation and system, with the right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder, $$\text {BMO}$$ BMO and $${{\,\mathrm{VMO}\,}}$$ VMO spaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when $$p=2$$ p = 2 , and hence the differential operator is the plain Laplacian. Yet in this classical linear setting, our contribution completes and augments the celebrated Schauder theory in Hölder spaces. A distinctive trait of our results is their sharpness, which is demonstrated by a family of apropos examples.

scholarly journals Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L1

2018 ◽  
Vol 2018 ◽  
pp. 1-15
Author(s):  
Abdeluaab Lidouh ◽  
Rachid Messaoudi
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Divergence Form ◽  
Second Order ◽  
Discrete Problem ◽  
Renormalized Solution ◽  
Order Linear ◽  
Right Hand ◽  
The Right

We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in L∞Ω and the right-hand side belongs to L1Ω; we extend the results where the case of linear finite elements approximation is considered. We prove that the unique solution of the discrete problem converges in W01,qΩ for every q with 1≤q<d/d-1 (d=2 or d=3) to the unique renormalized solution of the problem. Statements and proofs remain valid in our case, which permits obtaining a weaker result when the right-hand side is a bounded Radon measure and, when the coefficients are smooth, an error estimate in W01,qΩ when the right-hand side f belongs to LrΩ verifying Tkf∈H1Ω for every k>0, for some r>1.

scholarly journals Hardy potential versus lower order terms in Dirichlet problems: regularizing effects

2022 ◽  
Vol 5 (1) ◽  
pp. 1-14
Author(s):  
David Arcoya ◽  
◽  
Lucio Boccardo ◽  
Luigi Orsina ◽  
◽  
...  
Keyword(s):  
Lower Order ◽  
Potential Versus ◽  
Dirichlet Problems ◽  
Hardy Potential ◽  
Right Hand ◽  
The Right ◽  
Regularizing Effects ◽  
Lower Order Terms

<abstract><p>In this paper, dedicated to Ireneo Peral, we study the regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy potentials in the right hand side.</p></abstract>

Regularity of plate equations with control concentrated in interior curves

1997 ◽  
Vol 127 (5) ◽  
pp. 1005-1025 ◽  
Author(s):  
Stéphane Jaffard ◽  
Marius Tucsnak
Keyword(s):  
Boundary Value Problems ◽  
Piezoelectric Actuator ◽  
Boundary Value ◽  
Piezoelectric Actuators ◽  
Function Spaces ◽  
Right Hand ◽  
Regularity Properties ◽  
Plate Equations ◽  
Regularity Results ◽  
The Right

SynopsisWe consider initial and boundary-value problems modelling the vibration of a plate with piezoelectric actuator. The usual models lead to the Bernoulli–Euler and Kirchhoff plate equations with right-hand side given by a distribution concentrated in an interior curve. We obtain regularity results which are stronger than those obtained by simply using the Sobolev regularity of the right-hand side. By duality, we obtain new trace regularity properties for the solutions of plate equations. Our results provide appropriate function spaces for the control of plates provided with piezoelectric actuators.

scholarly journals On solving non-homogeneous partial differential equations with right-hand side defined on the grid

2021 ◽  
Vol 31 (3) ◽  
pp. 443-457
Author(s):  
L.I. Rubina ◽  
O.N. Ul'yanov
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Grid Point ◽  
Original Equation ◽  
Partial Differential ◽  
First Order ◽  
Ode System ◽  
Right Hand ◽  
The Right

An algorithm is proposed for obtaining solutions of partial differential equations with right-hand side defined on the grid $\{ x_{1}^{\mu}, x_{2}^{\mu}, \ldots, x_{n}^{\mu}\},\ (\mu=1,2,\ldots,N)\colon f_{\mu}=f(x_{1}^{\mu}, x_{2}^{\mu}, \ldots, x_{n}^{\mu}).$ Here $n$ is the number of independent variables in the original partial differential equation, $N$ is the number of rows in the grid for the right-hand side, $f=f( x_{1}, x_{2}, \ldots, x_{n})$ is the right-hand of the original equation. The algorithm implements a reduction of the original equation to a system of ordinary differential equations (ODE system) with initial conditions at each grid point and includes the following sequence of actions. We seek a solution to the original equation, depending on one independent variable. The original equation is associated with a certain system of relations containing arbitrary functions and including the partial differential equation of the first order. For an equation of the first order, an extended system of equations of characteristics is written. Adding to it the remaining relations containing arbitrary functions, and demanding that these relations be the first integrals of the extended system of equations of characteristics, we arrive at the desired ODE system, completing the reduction. The proposed algorithm allows at each grid point to find a solution of the original partial differential equation that satisfies the given initial and boundary conditions. The algorithm is used to obtain solutions of the Poisson equation and the equation of unsteady axisymmetric filtering at the points of the grid on which the right-hand sides of the corresponding equations are given.

scholarly journals Nonresonance conditions and extremal solutions for first-order impulsive problems under weak assumptions

2003 ◽  
Vol 44 (3) ◽  
pp. 393-407 ◽  
Author(s):  
Daniel Franco ◽  
Rodrigo L. Pouso
Keyword(s):  
Boundary Conditions ◽  
Extremal Solutions ◽  
First Order ◽  
Weak Regularity ◽  
Right Hand ◽  
Impulsive Problems ◽  
Functional Boundary ◽  
Functional Boundary Conditions ◽  
The Right ◽  
Complete Characterisation

AbstractIn this work we shall study the existence of extremal solutions for an impulsive problem with functional-boundary conditions and weak regularity assumptions, not only on the right-hand side of the equation and on the functions that define the boundary conditions, but also on the impulse functions, which will be required to be nondecreasing, but not continuous as well, as is customary in the literature.Moreover, in order to prove one of our results we shall study a general impulsive linearproblem, giving a complete characterisation of resonance for it.

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

Construct Validity

2017 ◽  
Author(s):  
Richard McCleary ◽  
David McDowall ◽  
Bradley J. Bartos
Keyword(s):  
Time Series ◽  
Construct Validity ◽  
Hypothesis Test ◽  
Secondary Data ◽  
Time Frame ◽  
Primary Data ◽  
Optimal Time ◽  
Intervention Model ◽  
Right Hand ◽  
The Right

Chapter 8 focuses on threats to construct validity arising from the left-hand side time series and the right-hand side intervention model. Construct validity is limited to questions of whether an observed effect can be generalized to alternative cause and effect measures. The “talking out” self-injurious behavior time series, shown in Chapter 5, are examples of primary data. Researchers often have no choice but to use secondary data that were collected by third parties for purposes unrelated to any hypothesis test. Even in those less-than-ideal instances, however, an optimal time series can be constructed by limiting the time frame and otherwise paying attention to regime changes. Threats to construct validity that arise from the right-hand side intervention model, such as fuzzy or unclear onset and responses, are controlled by paying close attention to the underlying theory. Even a minimal theory should specify the onset and duration of an impact.

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