### Abstract

We study the comparison principle and interior Hölder continuity of viscosity solutions of F(x, u(x),Du(x),D^{2}u(x)) + H(x,Du(x))-f(x) = 0 in O, where F satisfies the standard "structure condition" and H has superlinear growth with respect to Du. Following Caffarelli, Crandall, Kocan and Świȩch [3], we first present the comparison principle between L^{p}-viscosity subsolution and L^{p}-strong supersolutions. We next show the interior Hölder continuity for L^{p}-viscosity solutions of the above equation. For this purpose, modifying some arguments in [1] by Caffarelli, we obtain the Harnack inequality for them when the growth order of H with respect to Du is less than 2.

Original language | English |
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Pages (from-to) | 493-512 |

Number of pages | 20 |

Journal | Advances in Differential Equations |

Volume | 7 |

Issue number | 4 |

Publication status | Published - 2002 Dec 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Advances in Differential Equations*,

*7*(4), 493-512.

**Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients.** / Koike, Shigeaki; Takahashi, Toshimi.

Research output: Contribution to journal › Article

*Advances in Differential Equations*, vol. 7, no. 4, pp. 493-512.

}

TY - JOUR

T1 - Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients

AU - Koike, Shigeaki

AU - Takahashi, Toshimi

PY - 2002/12/1

Y1 - 2002/12/1

N2 - We study the comparison principle and interior Hölder continuity of viscosity solutions of F(x, u(x),Du(x),D2u(x)) + H(x,Du(x))-f(x) = 0 in O, where F satisfies the standard "structure condition" and H has superlinear growth with respect to Du. Following Caffarelli, Crandall, Kocan and Świȩch [3], we first present the comparison principle between Lp-viscosity subsolution and Lp-strong supersolutions. We next show the interior Hölder continuity for Lp-viscosity solutions of the above equation. For this purpose, modifying some arguments in [1] by Caffarelli, we obtain the Harnack inequality for them when the growth order of H with respect to Du is less than 2.

AB - We study the comparison principle and interior Hölder continuity of viscosity solutions of F(x, u(x),Du(x),D2u(x)) + H(x,Du(x))-f(x) = 0 in O, where F satisfies the standard "structure condition" and H has superlinear growth with respect to Du. Following Caffarelli, Crandall, Kocan and Świȩch [3], we first present the comparison principle between Lp-viscosity subsolution and Lp-strong supersolutions. We next show the interior Hölder continuity for Lp-viscosity solutions of the above equation. For this purpose, modifying some arguments in [1] by Caffarelli, we obtain the Harnack inequality for them when the growth order of H with respect to Du is less than 2.

UR - http://www.scopus.com/inward/record.url?scp=12744266482&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=12744266482&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:12744266482

VL - 7

SP - 493

EP - 512

JO - Advances in Differential Equations

JF - Advances in Differential Equations

SN - 1079-9389

IS - 4

ER -