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.

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

2008 ◽  
Vol 15 (3) ◽  
pp. 555-569
Author(s):  
Tariel Kiguradze
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Hyperbolic Equation ◽  
Well Posedness ◽  
Nonlinear Hyperbolic Equations ◽  
Right Hand

Abstract In the rectangle Ω = [0, a] × [0, b] the nonlinear hyperbolic equation 𝑢(2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

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.

A Method for the Numerical Solution of a Boundary Value Problem for a Linear Differential Equation with Interval Parameters

2019 ◽  
Vol 16 (07) ◽  
pp. 1850115 ◽  
Author(s):  
Nizami A. Gasilov ◽  
Müjdat Kaya
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Numerical Method ◽  
Boundary Value ◽  
Real Life ◽  
Boundary Values ◽  
Right Hand ◽  
Linear Differential ◽  
The Right

In many real life applications, the behavior of the system is modeled by a boundary value problem (BVP) for a linear differential equation. If the uncertainties in the boundary values, the right-hand side function and the coefficient functions are to be taken into account, then in many cases an interval boundary value problem (IBVP) arises. In this study, for such an IBVP, we propose a different approach than the ones in common use. In the investigated IBVP, the boundary values are intervals. In addition, we model the right-hand side and coefficient functions as bunches of real functions. Then, we seek the solution of the problem as a bunch of functions. We interpret the IBVP as a set of classical BVPs. Such a classical BVP is constructed by taking a real number from each boundary interval, and a real function from each bunch. We define the bunch consisting of the solutions of all the classical BVPs to be the solution of the IBVP. In this context, we develop a numerical method to obtain the solution. We reduce the complexity of the method from [Formula: see text] to [Formula: see text] through our analysis. We demonstrate the effectiveness of the proposed approach and the numerical method by test examples.

scholarly journals Second order elliptic boundary value problems in spaces with homogeneous norms

1980 ◽  
Vol 29 (1) ◽  
pp. 1-13 ◽  
Author(s):  
A. J. Pryde
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Partial Differential Operator ◽  
Second Order ◽  
Elliptic Boundary Value ◽  
First Order ◽  
Constant Coefficients ◽  
Regularity Results ◽  
Homogeneous Norms

AbstractWe consider the interior and Dirichiet problems and problems with first order boundary conditions, for a second order homogeneous elliptic partial differential operator with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. From there we obtain a priori estimates and regularity results for boundary value problems in Sobolev spaces.

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