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On the isoperimetric problem with perimeter density $r^p$
An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators
Department of Mathematics, Korea University, 1 Anam-dong, Sungbuk-gu, Seoul, 136-701, Republic of Korea |
$u$ |
$L_p\left( (0, T); Λ_{γ+m}\right)$ |
$\begin{align}\notag&\frac{\partial u}{\partial t}(t, x) = ψ(t, i\nabla)u(t, x)+f(t, x), \;\;\;(t, x) ∈ (0, T) × {\bf{R}}^d \\& u(0, x) = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)\end{align}$ |
$d ∈ \mathbb{N}$ |
$p ∈ (1, ∞]$ |
$γ, m ∈ (0, ∞)$ |
$Λ_{γ+m}$ |
${\bf{R}}^d$ |
$γ+m$ |
$f ∈ L_p\left( (0, T) ; Λ_{γ} \right)$ |
$ψ(t, i\nabla)$ |
$ψ(t, ξ)$ |
$ψ(t, i\nabla)u(t, x) = \mathcal{F}^{-1}[ψ(t, ξ){\mathcal{F}}\left[u(t, ·)\right]\left(ξ)\right](x), $ |
$\begin{align*}\Re[ψ(t, ξ)] ≤ -ν|ξ|^{γ}, \end{align*}$ |
$\begin{align*}|D_{ξ}^{α}ψ(t, ξ)|≤ν^{-1}|ξ|^{γ-|α|}.\end{align*}$ |
$\begin{align}\int_0^T \|u(t, ·)\|^p_{Λ_{γ+m}} dt ≤ N \int_0^T \|f(t, ·)\|^p_{Λ_{m}} dt, \;\;\;\;\;\;\;\;\;\;(2)\end{align}$ |
$N$ |
$d$ |
$p$ |
$γ$ |
$ν$ |
$m$ |
$T$ |
$L_p$ |
$f ∈ L_p((0, T);C^{n+α})$ |
$u ∈ L_p((0, T);C^{γ+n+α}({\bf{R}}^d))$ |
$u$ |
$\begin{align}\int_0^T \|u(t, ·)\|^p_{C^{γ+n+α}}dt ≤N \int_0^T \|f(t, ·)\|^p_{C^{n+α}}dt, \;\;\;\;\;\;\;\;\;\;(3)\end{align}$ |
$n ∈ \mathbb{Z}_+$ |
$α ∈ (0, 1)$ |
$γ+α \notin \mathbb{Z}_+$ |
References:
[1] |
H. Abels, Pseudodifferential and Singular Integral Operators: An Introduction with Applications, Walter de Gruyter, 2012. |
[2] |
H. Dong and S. Kim,
Partial schauder estimates for second-order elliptic and parabolic equations, Calculus of Variations and Partial Differential Equations, 40 (2011), 481-500.
doi: 10.1007/s00526-010-0348-9. |
[3] |
H. Dong and S. Kim, Partial schauder estimates for second-order elliptic and parabolic equations: a revisit, arXiv: 1502.00886, 2015.
doi: 10.1007/s00526-010-0348-9. |
[4] |
L. Grafakos, Classical Fourier Analysis, volume 249, Springer, 2008. |
[5] |
L. Grafakos, Modern Fourier Analysis, volume 250, Springer, 2009.
doi: 10.1007/978-0-387-09434-2. |
[6] |
L. Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-differential Operators, volume 274, Springer Science & Business Media, 2007.
doi: 10.1007/978-3-540-49938-1. |
[7] |
N. Jacob, Pseudo-Differential Operators & Markov Processes: Generators and Their Potential Theory, volume 2, Imperial College Press, 2002.
doi: 10.1142/9781860949562. |
[8] |
I. Kim, K.-H. Kim and S. Lim,
Parabolic BMO estimates for pseudo-differential operators of arbitrary order, Journal of Mathematical Analysis and Applications, 427 (2015), 557-580.
doi: 10.1016/j.jmaa.2015.02.065. |
[9] |
I. Kim, S. Lim and K.-H. Kim,
An Lq(Lp)-theory for parabolic pseudo-differential equations: Calderón-Zygmund approach,, Potential Analysis, (2016), 1-21.
doi: 10.1007/s11118-016-9552-3. |
[10] |
N. V. Krylov,
The Calderón-Zygmund theorem and parabolic equations in $ {L_p}\left({\mathbb{R},{C^{2 + \alpha }}} \right) $-spaces, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 1 (2002), 799-820.
|
[11] |
N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, volume 96, American Mathematical Society Providence, RI, 2008.
doi: 10.1090/gsm/096. |
[12] |
Y. Lin and S.Z. Lu,
Pseudo-differential operators on Sobolev and Lipschitz spaces, Acta Mathematica Sinica, English Series, 16 (2010), 131-142.
doi: 10.1007/s10114-010-8109-4. |
[13] |
L. Lorenzi,
Optimal Schauder estimates for parabolic problems with data measurable with respect to time, SIAM Journal on Mathematical Analysis, 31 (2000), 588-615.
doi: 10.1137/S0036141098342842. |
[14] |
R. Mikulevičius and H. Pragarauskas,
On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces, Lithuanian Mathematical Journal, 32 (1992), 238-264.
doi: 10.1007/BF02450422. |
[15] |
R. Mikulevicius and H. Pragarauskas,
On the cauchy problem for integro-differential operators in hölder classes and the uniqueness of the martingale problem, Potential Analysis, 40 (2014), 539-563.
doi: 10.1007/s11118-013-9359-4. |
[16] |
E. M. Stein and T. S. Murphy, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, volume 3, Princeton University Press, 1993. |
show all references
References:
[1] |
H. Abels, Pseudodifferential and Singular Integral Operators: An Introduction with Applications, Walter de Gruyter, 2012. |
[2] |
H. Dong and S. Kim,
Partial schauder estimates for second-order elliptic and parabolic equations, Calculus of Variations and Partial Differential Equations, 40 (2011), 481-500.
doi: 10.1007/s00526-010-0348-9. |
[3] |
H. Dong and S. Kim, Partial schauder estimates for second-order elliptic and parabolic equations: a revisit, arXiv: 1502.00886, 2015.
doi: 10.1007/s00526-010-0348-9. |
[4] |
L. Grafakos, Classical Fourier Analysis, volume 249, Springer, 2008. |
[5] |
L. Grafakos, Modern Fourier Analysis, volume 250, Springer, 2009.
doi: 10.1007/978-0-387-09434-2. |
[6] |
L. Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-differential Operators, volume 274, Springer Science & Business Media, 2007.
doi: 10.1007/978-3-540-49938-1. |
[7] |
N. Jacob, Pseudo-Differential Operators & Markov Processes: Generators and Their Potential Theory, volume 2, Imperial College Press, 2002.
doi: 10.1142/9781860949562. |
[8] |
I. Kim, K.-H. Kim and S. Lim,
Parabolic BMO estimates for pseudo-differential operators of arbitrary order, Journal of Mathematical Analysis and Applications, 427 (2015), 557-580.
doi: 10.1016/j.jmaa.2015.02.065. |
[9] |
I. Kim, S. Lim and K.-H. Kim,
An Lq(Lp)-theory for parabolic pseudo-differential equations: Calderón-Zygmund approach,, Potential Analysis, (2016), 1-21.
doi: 10.1007/s11118-016-9552-3. |
[10] |
N. V. Krylov,
The Calderón-Zygmund theorem and parabolic equations in $ {L_p}\left({\mathbb{R},{C^{2 + \alpha }}} \right) $-spaces, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 1 (2002), 799-820.
|
[11] |
N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, volume 96, American Mathematical Society Providence, RI, 2008.
doi: 10.1090/gsm/096. |
[12] |
Y. Lin and S.Z. Lu,
Pseudo-differential operators on Sobolev and Lipschitz spaces, Acta Mathematica Sinica, English Series, 16 (2010), 131-142.
doi: 10.1007/s10114-010-8109-4. |
[13] |
L. Lorenzi,
Optimal Schauder estimates for parabolic problems with data measurable with respect to time, SIAM Journal on Mathematical Analysis, 31 (2000), 588-615.
doi: 10.1137/S0036141098342842. |
[14] |
R. Mikulevičius and H. Pragarauskas,
On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces, Lithuanian Mathematical Journal, 32 (1992), 238-264.
doi: 10.1007/BF02450422. |
[15] |
R. Mikulevicius and H. Pragarauskas,
On the cauchy problem for integro-differential operators in hölder classes and the uniqueness of the martingale problem, Potential Analysis, 40 (2014), 539-563.
doi: 10.1007/s11118-013-9359-4. |
[16] |
E. M. Stein and T. S. Murphy, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, volume 3, Princeton University Press, 1993. |
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