# General least gradient problems with obstacle

• Morteza Fotouhi
Article

## Abstract

We study existence, structure, uniqueness and regularity of solutions of the obstacle problem
\begin{aligned} \inf _{u\in BV_f(\Omega )}\int _{\Omega }\phi (x,Du), \end{aligned}
where $$BV_f(\Omega )=\{u\in BV({\mathbb {R}}^n): u\ge \psi \text { in }\Omega \text { and } u|_{\partial \Omega }=f|_{\partial \Omega }\}$$, $$f \in W^{1,1}_0({\mathbb {R}}^n)$$, $$\psi$$ is the obstacle, and $$\phi (x,\xi )$$ is a convex, continuous and homogeneous function of degree one with respect to the $$\xi$$ variable. We show that every minimizer of this problem is also a minimizer of the least gradient problem
\begin{aligned} \inf _{u\in {\mathcal {A}}_f(\Omega )}\int _{{\mathbb {R}}^n}\phi (x,Du), \end{aligned}
where $${\mathcal {A}}_f(\Omega )=\{u\in BV(\Omega ): u\ge \psi , \text { and } u=f \text { in }\Omega ^c\}$$. Moreover, there exists a vector field T with $$\nabla \cdot T \le 0$$ in $$\Omega$$ which determines the structure of all minimizers of these two problems, and T is divergence free on $$\{x\in \Omega : u(x)>\psi (x)\}$$ for any minimizer u. We also present uniqueness and regularity results that are based on maximum principles for minimal surfaces. Since minimizers of the least gradient problems with obstacle do not hit small enough obstacles, the results presented in this paper extend several results in the literature about least gradient problems without obstacle.

## Mathematics Subject Classification

35B65 35R35 49N60

## References

1. 1.
Alberti, G.: A Lusin type theorem for gradients. J. Funct. Anal. 100(1), 110–118 (1991)
2. 2.
Amar, M., Bellettini, G.: A notion of total variation depending on a metric with discontinuous coefficients. Ann. Inst. Henri Poincare Anal. Non Lineaire 11, 91–133 (1994)
3. 3.
Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. (4) 135(1), 293–318 (1983)
4. 4.
Bombieri, E., De Giorgi, E., Giusti, E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)
5. 5.
Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. North-Holland-Elsevier, Amsterdam (1976)
6. 6.
Górny, W.: Planar least gradient problem: existence, regularity and anisotropic case. Calc. Var. Partial Differ. Equ. 57(4), 27 (2018). Art. 98
7. 7.
Jerrard, R.L., Moradifam, A., Nachman, A.I.: Existence and uniqueness of minimizers of general least gradient problems. J. Reine Angew. Math. 734, 71–97 (2018)
8. 8.
Juutinen, P.: p-Harmonic approximation of functions of least gradient. Indiana Univ. Math. J. 54, 1015–1029 (2005)
9. 9.
Kohn, R., Strang, G.: The Constrained Least Gradient Problem. Nonclassical Continuum Mechanics, (Durham, 1986). London Mathematical Society Lecture Note Series, vol. 122, pp. 226–243. Cambridge University Press, Cambridge (1987)Google Scholar
10. 10.
Mazón, J.M., Rossi, J.D., Segura de León, S.: Functions of least gradient and 1-harmonic functions. Indiana Univ. Math. J. 63(4), 1067–1084 (2014)
11. 11.
Miranda, M.: Comportamento delle successioni convergenti di frontiere minimali. Rend. Sem. Mat. Univ. Padova 38, 238–257 (1967)
12. 12.
Moradifam, A.: Existence and structure of minimizers of least gradient problems. Indiana Univ. Math. J. 67(3), 1025–1037 (2018)
13. 13.
Moradifam, A.: Least gradient problems with Neumann boundary condition. J. Differ. Equ. 263(11), 7900–7918 (2017)
14. 14.
Moradifam, A., Nachman, A., Tamasan, A.: Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions. SIAM J. Math. Anal. 44, 3969–3990 (2012)
15. 15.
Moradifam, A., Nachman, A., Timonov, A.: A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data. Inverse Probl. 28, 084003 (2012). (23pp)
16. 16.
Nachman, A., Tamasan, A., Timonov, A.: Conductivity imaging with a single measurement of boundary and interior data. Inverse Probl. 23, 2551–2563 (2007)
17. 17.
Nachman, A., Tamasan, A., Timonov, A.: Recovering the conductivity from a single measurement of interior data. Inverse Probl. 25, 035014 (2009). (16pp)
18. 18.
Nachman, A., Tamasan, A., Timonov, A.: Reconstruction of planar conductivities in subdomains from incomplete data. SIAM J. Appl. Math. 70(8), 3342–3362 (2010)
19. 19.
Schoen, R., Simon Jr., L., Almgren, F.J.: Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. Acta Math. 139(3–4), 217–265 (1977)
20. 20.
Simon, L.: A strict maximum principle for area minimizing hypersurfaces. J. Differ. Geom. 26(2), 327–335 (1987)
21. 21.
Sternberg, P., Williams, G., Ziemer, W.P.: Existence, uniqueness, and regularity for functions of least gradient. J. Reine Angew. Math. 430, 35–60 (1992)
22. 22.
Sternberg, P., Williams, G., Ziemer, W.P.: The constrained least gradient problem in $$R^n$$. Trans. Am. Math. Soc. 339(1), 403–432 (1993)
23. 23.
Ziemer, W.P., Zumbrun, K.: The obstacle problem for functions of least gradient. Math. Bohem. 124(2–3), 193–219 (1999)
24. 24.
Zuniga, A.: Continuity of minimizers to weighted least gradient problems. Nonlinear Anal. 178, 86–109 (2019)