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Large time behavior of solutions of the heat equation with inverse square potential
The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4
Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via R. Cozzi 53, I-20125 Milano, Italy |
$\int_{Ω} [L(\nabla v(x))+g(x, v(x))]dx~~~ \hbox {on}~~~ u^0+W^{1, p}_0(Ω)$ |
$\Omega\subset \mathbb R^N$ |
$L(ξ) = l(|ξ|) = \frac{1}{p}|ξ|^p$ |
$ u^0∈ W^{1, p}(Ω)$ |
${\rm div }(|\nabla u|^ {p-2}\nabla u) = f.$ |
$3≤ p < 4$ |
$g$ |
$ u^*$ |
$\nabla u^*∈ W^{s, 2}_{loc}(\Omega)$ |
$0 < s < 4-p$ |
$p = 3$ |
$u^*$ |
$\nabla u^*∈ W^{s, 2}_{loc}(\Omega)$ |
$s < 1$ |
$N$ |
$N = 1$ |
$p = 3$ |
$u$ |
$\nabla u^*$ |
$W^{1, 2}_{loc}(\Omega)$ |
References:
[1] |
B. Avelin, T. Kuusi and G. Mingione,
Nonlinear Calderon-Zygmund theory in the limiting case, Arch. Rat. Mech. Anal., 227 (2018), 663-714.
doi: 10.1007/s00205-017-1171-7. |
[2] |
A. Cellina,
The regularity of solutions to some variational problems, including the p-Laplace equation for 2 ≤ p < 3, ESAIM: COCV, 23 (2017), 1543-1553.
doi: 10.1051/cocv/2016064. |
[3] |
A. Cianchi and V. G. Maz'ya,
Second-order two-sided estimates in nonlinear elliptic problems, Archive for Rational Mechanics and Analysis, (2017), 1-31.
doi: 10.1007/s00205-018-1223-7. |
[4] |
F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, Heidelberg, 2012.
doi: 10.1007/978-1-4471-2807-6. |
[5] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[6] |
L. Esposito and G. Mingione,
Some remarks on the regularity of weak solutions of degenerate elliptic systems, Rev. Mat. Complutense, 11 (1998), 203-219.
|
[7] |
E. Giusti, Metodi Diretti Nel Calcolo Delle Variazioni, Unione Matematica Italiana, Bologna, 1994. |
[8] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian. Academic Press, New York-London, 1968. |
[9] |
J. J. Manfredi and A. Weitsman,
On the Fatou Theorem for p-harmonic functions, Comm. Partial Differential Equations, 13 (1988), 651-668.
doi: 10.1080/03605308808820556. |
[10] |
W. P. Ziemer, Weakly Differentiable Functions, Springer, Berlin, 1989.
doi: 10.1007/978-1-4612-1015-3. |
show all references
References:
[1] |
B. Avelin, T. Kuusi and G. Mingione,
Nonlinear Calderon-Zygmund theory in the limiting case, Arch. Rat. Mech. Anal., 227 (2018), 663-714.
doi: 10.1007/s00205-017-1171-7. |
[2] |
A. Cellina,
The regularity of solutions to some variational problems, including the p-Laplace equation for 2 ≤ p < 3, ESAIM: COCV, 23 (2017), 1543-1553.
doi: 10.1051/cocv/2016064. |
[3] |
A. Cianchi and V. G. Maz'ya,
Second-order two-sided estimates in nonlinear elliptic problems, Archive for Rational Mechanics and Analysis, (2017), 1-31.
doi: 10.1007/s00205-018-1223-7. |
[4] |
F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, Heidelberg, 2012.
doi: 10.1007/978-1-4471-2807-6. |
[5] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[6] |
L. Esposito and G. Mingione,
Some remarks on the regularity of weak solutions of degenerate elliptic systems, Rev. Mat. Complutense, 11 (1998), 203-219.
|
[7] |
E. Giusti, Metodi Diretti Nel Calcolo Delle Variazioni, Unione Matematica Italiana, Bologna, 1994. |
[8] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian. Academic Press, New York-London, 1968. |
[9] |
J. J. Manfredi and A. Weitsman,
On the Fatou Theorem for p-harmonic functions, Comm. Partial Differential Equations, 13 (1988), 651-668.
doi: 10.1080/03605308808820556. |
[10] |
W. P. Ziemer, Weakly Differentiable Functions, Springer, Berlin, 1989.
doi: 10.1007/978-1-4612-1015-3. |
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