A complex with properties of uniform ellipticity

2021 ◽  
Vol 85 ◽  
Author(s):  
Il'ya Anatol'evich Ivanov-Pogodaev ◽  
Aleksei Yakovlevich Kanel-Belov
Keyword(s):  
Uniform Ellipticity

scholarly journals A Regularity Result for the p-Laplacian Near Uniform Ellipticity

2016 ◽  
Vol 48 (3) ◽  
pp. 2059-2075 ◽  
Author(s):  
Carlo Mercuri ◽  
Giuseppe Riey ◽  
Berardino Sciunzi
Keyword(s):  
Regularity Result ◽  
Uniform Ellipticity

scholarly journals Uniform ellipticity and (p,q) growth

2020 ◽  
pp. 124451
Author(s):  
Cristiana De Filippis ◽  
Francesco Leonetti
Keyword(s):  
Uniform Ellipticity

scholarly journals TWO-PARAMETER UNIFORMLY ELLIPTIC STURM–LIOUVILLE PROBLEMS WITH EIGENPARAMETER-DEPENDENT BOUNDARY CONDITIONS

2005 ◽  
Vol 48 (3) ◽  
pp. 531-547 ◽  
Author(s):  
Bhattacharyya T. Bhattacharyya ◽  
Mohandas J. P. Mohandas
Keyword(s):  
Boundary Conditions ◽  
Real Numbers ◽  
Uniform Ellipticity ◽  
Sturm Liouville ◽  
Two Parameter

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.

scholarly journals Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Grégory Faye ◽  
Thomas Giletti ◽  
Matt Holzer
Keyword(s):  
Diffusion Coefficient ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Spreading Speed ◽  
Reaction Diffusion Equation ◽  
Unstable State ◽  
Homogeneous State ◽  
Ellipticity Condition ◽  
Uniform Ellipticity ◽  
Asymptotic Spreading

<p style='text-indent:20px;'>We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotonicity of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems at <inline-formula><tex-math id="M1">\begin{document}$ x = \pm \infty $\end{document}</tex-math></inline-formula>. Most notably, when the profile of the diffusion coefficient is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the diffusion coefficient is decreasing and in the regime where the forcing speed is precisely the selected spreading speed.</p>

A note on positivity of two-dimensional differential operators

Filomat ◽  
2017 ◽  
Vol 31 (14) ◽  
pp. 4651-4663 ◽  
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.

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

2011 ◽  
Vol 2011 ◽  
pp. 1-35 ◽  
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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