Nonlinear Operators with Discontinuous Coefficients: Parabolic Systems

2003 ◽  
pp. 217-234
Keyword(s):  
Parabolic Systems ◽  
Discontinuous Coefficients ◽  
Nonlinear Operators

Higher differentiability of solutions of parabolic systems with discontinuous coefficients

2016 ◽  
Vol 94 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Flavia Giannetti ◽  
Antonia Passarelli di Napoli ◽  
Christoph Scheven
Keyword(s):  
Parabolic Systems ◽  
Discontinuous Coefficients ◽  
Differentiability Of Solutions ◽  
Higher Differentiability

On higher differentiability of solutions of parabolic systems with discontinuous coefficients and (p, q)-growth

2019 ◽  
Vol 150 (1) ◽  
pp. 419-451
Author(s):  
Flavia Giannetti ◽  
Antonia Passarelli di Napoli ◽  
Christoph Scheven
Keyword(s):  
Weak Solutions ◽  
A Priori ◽  
A Priori Estimate ◽  
Parabolic Systems ◽  
Discontinuous Coefficients ◽  
Sobolev Type ◽  
Dirichlet Problems ◽  
Priori Estimate ◽  
Image Position ◽  
Higher Differentiability

AbstractWe consider weak solutions $u:\Omega _T\to {\open R}^N$ to parabolic systems of the type $$u_t-{\rm div}\;a(x,t,Du) = 0\quad {\rm in}\;\Omega _T = \Omega \times (0,T),$$where the function a(x, t, ξ) satisfies (p, q)-growth conditions. We give an a priori estimate for weak solutions in the case of possibly discontinuous coefficients. More precisely, the partial maps $x\mapsto a(x,t,\xi )$ under consideration may not be continuous, but may only possess a Sobolev-type regularity. In a certain sense, our assumption means that the weak derivatives $D_xa(\cdot ,\cdot ,\xi )$ are contained in the class $L^\alpha (0,T;L^\beta (\Omega ))$, where the integrability exponents $\alpha ,\beta $ are coupled by $$\displaystyle{{p(n + 2)-2n} \over {2\alpha }} + \displaystyle{n \over \beta } = 1-\kappa $$for some κ ∈ (0,1). For the gap between the two growth exponents we assume $$2 \les p < q \les p + \displaystyle{{2\kappa } \over {n + 2}}.$$Under further assumptions on the integrability of the spatial gradient, we prove a result on higher differentiability in space as well as the existence of a weak time derivative $u_t\in L^{p/(q-1)}_{{\rm loc}}(\Omega _T)$. We use the corresponding a priori estimate to deduce the existence of solutions of Cauchy–Dirichlet problems with the mentioned higher differentiability property.

Solvability of a 2 x 2 block operator matrix of chandrasekhar type on a Bananch algebra

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5169-5175 ◽  
Author(s):  
H.H.G. Hashem
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence Of Solutions ◽  
Banach Algebras ◽  
Operator Matrix ◽  
Nonlinear Operators ◽  
Block Operator Matrix ◽  
Block Operator ◽  
Quadratic Integral ◽  
Quadratic Integral Equations

In this paper, we study the existence of solutions for a system of quadratic integral equations of Chandrasekhar type by applying fixed point theorem of a 2 x 2 block operator matrix defined on a nonempty bounded closed convex subsets of Banach algebras where the entries are nonlinear operators.

scholarly journals A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

2021 ◽  
Author(s):  
Alessandro Goffi ◽  
Francesco Pediconi
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Strong Maximum Principle ◽  
Second Order ◽  
Fully Nonlinear Equations ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Second Order Equations ◽  
Curvature Type

AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

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