Uniform ellipticity and (p,q) growth

AbstractWe consider the two-parameter Sturm–Liouville system$$ -y_1''+q_1y_1=(\lambda r_{11}+\mu r_{12})y_1\quad\text{on }[0,1], $$with the boundary conditions$$ \frac{y_1'(0)}{y_1(0)}=\cot\alpha_1\quad\text{and}\quad\frac{y_1'(1)}{y_1(1)}=\frac{a_1\lambda+b_1}{c_1\lambda+d_1}, $$and$$ -y_2''+q_2y_2=(\lambda r_{21}+\mu r_{22})y_2\quad\text{on }[0,1], $$with the boundary conditions$$ \frac{y_2'(0)}{y_2(0)} =\cot\alpha_2\quad\text{and}\quad\frac{y_2'(1)}{y_2(1)}=\frac{a_2\mu+b_2}{c_2\mu+d_2}, $$subject to the uniform-left-definite and uniform-ellipticity conditions; where $q_{i}$ and $r_{ij}$ are continuous real valued functions on $[0,1]$, the angle $\alpha_{i}$ is in $[0,\pi)$ and $a_{i}$, $b_{i}$, $c_{i}$, $d_{i}$ are real numbers with $\delta_{i}=a_{i}d_{i}-b_{i}c_{i}>0$ and $c_{i}\neq0$ for $i,j=1,2$. Results are given on asymptotics, oscillation of eigenfunctions and location of eigenvalues.

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A complex with properties of uniform ellipticity

Izvestiya Mathematics ◽

10.1070/im8978 ◽

2021 ◽

Vol 85 ◽

Author(s):

Il'ya Anatol'evich Ivanov-Pogodaev ◽

Aleksei Yakovlevich Kanel-Belov

Keyword(s):

Uniform Ellipticity

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Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

The Annals of Probability ◽

10.1214/13-aop833 ◽

2014 ◽

Vol 42 (6) ◽

pp. 2558-2594 ◽

Cited By ~ 7

Author(s):

Scott N. Armstrong ◽

Charles K. Smart

Keyword(s):

Nonlinear Equations ◽

Stochastic Homogenization ◽

Fully Nonlinear Equations ◽

Fully Nonlinear ◽

Uniform Ellipticity

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A note on positivity of two-dimensional differential operators

Filomat ◽

10.2298/fil1714651a ◽

2017 ◽

Vol 31 (14) ◽

pp. 4651-4663 ◽

Cited By ~ 1

Author(s):

Allaberen Ashyralyev ◽

Sema Akturk

Keyword(s):

Boundary Condition ◽

Differential Operator ◽

Half Plane ◽

Differential Operators ◽

Elliptic Problems ◽

Well Posedness ◽

Two Dimensional ◽

Ellipticity Condition ◽

Fractional Spaces ◽

Uniform Ellipticity

We consider the two-dimensional differential operator A(t,x)u(t,x) = -a11 (t, x) utt -a22(t,x)uxx +?u defined on functions on the half-plane R+ x R with the boundary condition u(0,x) = 0, x ? R where aii(t,x), i = 1,2 are continuously differentiable and satisfy the uniform ellipticity condition a2 11(t,x) + a222(t,x)? ? > 0, ? > 0. The structure of fractional spaces E?,1 (L1 (R+ x R), A(t,x)) generated by the operator A(t,x) is investigated. The positivity of A(t,x) in L1 (W2?1(R+ x R)) spaces is established. In applications, theorems on well-posedness in L1 (W2?1 (R+ x R)) spaces of elliptic problems are obtained.

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The Cauchy-Dirichlet Problem for a Class of Linear Parabolic Differential Equations with Unbounded Coefficients in an Unbounded Domain

International Journal of Stochastic Analysis ◽

10.1155/2011/469806 ◽

2011 ◽

Vol 2011 ◽

pp. 1-35 ◽

Cited By ~ 3

Author(s):

Gerardo Rubio

Keyword(s):

Dirichlet Problem ◽

Differential Equations ◽

Parabolic Partial Differential Equations ◽

Bounded Domains ◽

Regular Boundary ◽

Parabolic Differential Equations ◽

Unbounded Coefficients ◽

Connected Set ◽

Locally Lipschitz ◽

Uniform Ellipticity

We consider the Cauchy-Dirichlet problem in [0,∞)×D for a class of linear parabolic partial differential equations. We assume that D⊂ℝd is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains.

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