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ABSTRACT
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ABSTRACT
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ABSTRACT
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ABSTRACT We consider a class of ultraparabolic operators of the following type L=div(A(x,t)D)+〈x,BD〉-∂ t , where B is a constant matrix, A(z)=A T (z)≥0. We show that if u is a solution of Lu=0 on ℝ N ×]0,T[ and u(x,0)=0, then each of the... more
ABSTRACT We consider a class of ultraparabolic operators of the following type L=div(A(x,t)D)+〈x,BD〉-∂ t , where B is a constant matrix, A(z)=A T (z)≥0. We show that if u is a solution of Lu=0 on ℝ N ×]0,T[ and u(x,0)=0, then each of the following conditions: |u(x,t)| can be bounded (in some sense) by e c|x| 2 , or u≥0, implies u≡0. We use a technique which is well known in the classic parabolic case and which relies on some pointwise estimates of the fundamental solution of L. Next, we prove a representation theorem and a Fatou type theorem for non-negative solutions of Lu=0 in ℝ N ×]0,T[.
We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolicequations. These estimates are independent of the modulus of continuity of the coefficients and generalizethe classical upper bounds by... more
We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolicequations. These estimates are independent of the modulus of continuity of the coefficients and generalizethe classical upper bounds by Aronson for uniformly parabolic equations.
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ABSTRACT We are concerned with Gaussian lower bounds for positive solutions of a family of hypoelliptic partial differential equations on homogeneous Lie groups. We describe a method that relies on the repeated use of an invariant Harnack... more
ABSTRACT We are concerned with Gaussian lower bounds for positive solutions of a family of hypoelliptic partial differential equations on homogeneous Lie groups. We describe a method that relies on the repeated use of an invariant Harnack inequality and on a suitable optimal control problem. We also obtain an accurate Gaussian lower bound for the fundamental solution of Kolmogorov type operators on non-homogeneous groups.
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We consider the second order differential equation <img src="/fulltext-image.asp?format=htmlnonpaginated&src=V237547274021394_html\11118_2004_Article_257106_TeX2GIFIE1.gif" border="0" alt=" $$\sum\limits_{i,j =... more
We consider the second order differential equation <img src="/fulltext-image.asp?format=htmlnonpaginated&src=V237547274021394_html\11118_2004_Article_257106_TeX2GIFIE1.gif" border="0" alt=" $$\sum\limits_{i,j = 1}^{m_0 } {\partial _{x_i } (a_{i,j} (x,t)\partial _{x_j } u) + \sum\limits_{i,j = 1}^N {b_{i,j} x_i \partial _{x_j } u - \partial _t u = \sum\limits_{j = 1}^{m_0 } {\partial _{x_j } F_j (x,t)} } } $$ " />, where (x,t)∈ℝN+1, 0m0⩽N, the coefficients ai,j belong to a suitable space of vanishing mean oscillation functions VMOL and B=(bi,j) is a constant real matrix. The aim of this paper is to study interior regularity for weak solutions to the above equation assuming that Fj belong to a function space of Morrey type.