# American Institute of Mathematical Sciences

March  2014, 9(1): 191-196. doi: 10.3934/nhm.2014.9.191

## A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function

 1 Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb

Received  May 2013 Revised  July 2013 Published  April 2014

The purpose of this note is to prove a version of the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. More precisely, let $\eta\in C^{0,\alpha}(\omega)$, $0<\alpha<1$ and let $\Omega_{\eta}$ be a domain which is locally subgraph of a function $\eta$. We prove that mapping $\gamma_{\eta}:u\mapsto u({\bf x},\eta({\bf x}))$ can be extended by continuity to a linear, continuous mapping from $H^1(\Omega_{\eta})$ to $H^s(\omega)$, $s<\alpha/2$. This study is motivated by analysis of fluid-structure interaction problems.
Citation: Boris Muha. A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. Networks & Heterogeneous Media, 2014, 9 (1) : 191-196. doi: 10.3934/nhm.2014.9.191
##### References:

show all references

##### References:
 [1] Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349 [2] Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255 [3] Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250 [4] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [5] Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020286 [6] Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365 [7] Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374 [8] Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315 [9] Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268 [10] Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362 [11] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [12] Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020285 [13] Shigui Ruan. Nonlinear dynamics in tumor-immune system interaction models with delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 541-602. doi: 10.3934/dcdsb.2020282 [14] Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395 [15] Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020280 [16] Xiaofeng Ren, David Shoup. The impact of the domain boundary on an inhibitory system: Interior discs and boundary half discs. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3957-3979. doi: 10.3934/dcds.2020048 [17] Wenrui Hao, King-Yeung Lam, Yuan Lou. Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 367-400. doi: 10.3934/dcdsb.2020283 [18] Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408 [19] Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260 [20] Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021021

2019 Impact Factor: 1.053