scholarly journals New Examples on Lavrentiev Gap Using Fractals

2020 ◽  
Vol 59 (5) ◽  
Author(s):  
Anna Kh. Balci ◽  
Lars Diening ◽  
Mikhail Surnachev
Keyword(s):  
Saddle Point ◽  
Variable Exponents ◽  
Smooth Functions ◽  
Double Phase

Abstract Zhikov showed 1986 with his famous checkerboard example that functionals with variable exponents can have a Lavrentiev gap. For this example it was crucial that the exponent had a saddle point whose value was exactly the dimension. In 1997 he extended this example to the setting of the double phase potential. Again it was important that the exponents crosses the dimensional threshold. Therefore, it was conjectured that the dimensional threshold plays an important role for the Lavrentiev gap. We show that this is not the case. Using fractals we present new examples for the Lavrentiev gap and non-density of smooth functions. We apply our method to the setting of variable exponents, the double phase potential and weighted p-energy.

Sobolev’s Inequality for Riesz Potentials of Functions in Musielak–Orlicz–Morrey Spaces Over Non-doubling Metric Measure Spaces

2019 ◽  
Vol 63 (2) ◽  
pp. 287-303
Author(s):  
Takao Ohno ◽  
Tetsu Shimomura
Keyword(s):  
Morrey Spaces ◽  
Variable Exponents ◽  
Metric Measure Spaces ◽  
Riesz Potentials ◽  
Sobolev’S Inequality ◽  
Double Phase ◽  
Measure Spaces

AbstractOur aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $I_{\unicode[STIX]{x1D6FC}(\,\cdot \,),\unicode[STIX]{x1D70F}}f$ of order $\unicode[STIX]{x1D6FC}(\,\cdot \,)$ with $f\in L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705},\unicode[STIX]{x1D703}}(X)$ over bounded non-doubling metric measure spaces. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

Campanato–Morrey spaces for the double phase functionals with variable exponents

2020 ◽  
Vol 197 ◽  
pp. 111827 ◽  
Author(s):  
Yoshihiro Mizuta ◽  
Eiichi Nakai ◽  
Takao Ohno ◽  
Tetsu Shimomura
Keyword(s):  
Morrey Spaces ◽  
Variable Exponents ◽  
Double Phase

Sobolev’s inequality for double phase functionals with variable exponents

2019 ◽  
Vol 31 (2) ◽  
pp. 517-527 ◽  
Author(s):  
Fumi-Yuki Maeda ◽  
Yoshihiro Mizuta ◽  
Takao Ohno ◽  
Tetsu Shimomura
Keyword(s):  
Variable Exponents ◽  
Hölder Continuous ◽  
Sobolev’S Inequality ◽  
Double Phase

AbstractOur aim in this paper is to establish generalizations of Sobolev’s inequality for double phase functionals {\Phi(x,t)=t^{p(x)}+a(x)t^{q(x)}}, where {p(\,{\cdot}\,)} and {q(\,{\cdot}\,)} satisfy log-Hölder conditions and {a(\,{\cdot}\,)} is nonnegative, bounded and Hölder continuous of order {\theta\in(0,1]}.

scholarly journals Regularity for minimizers for functionals of double phase with variable exponents

2019 ◽  
Vol 9 (1) ◽  
pp. 710-728 ◽  
Author(s):  
Maria Alessandra Ragusa ◽  
Atsushi Tachikawa
Keyword(s):  
Variable Exponents ◽  
Double Phase ◽  
Phase Type ◽  
Regularity Results

Abstract The functionals of double phase type $$\begin{array}{} \displaystyle {\cal H} (u):= \int \left(|Du|^{p} + a(x)|Du|^{q} \right) dx, ~~ ~~~(q \gt p \gt 1,~~a(x)\geq 0) \end{array}$$ are introduced in the epoch-making paper by Colombo-Mingione [1] for constants p and q, and investigated by them and Baroni. They obtained sharp regularity results for minimizers of such functionals. In this paper we treat the case that the exponents are functions of x and partly generalize their regularity results.

SOBOLEV’S INEQUALITY FOR MUSIELAK–ORLICZ–MORREY SPACES OVER METRIC MEASURE SPACES

2019 ◽  
pp. 1-15
Author(s):  
TAKAO OHNO ◽  
TETSU SHIMOMURA
Keyword(s):  
Morrey Spaces ◽  
Variable Exponents ◽  
Metric Measure Spaces ◽  
Riesz Potentials ◽  
Sobolev’S Inequality ◽  
Double Phase ◽  
Measure Spaces

Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $J_{\unicode[STIX]{x1D6FC}(\cdot )}^{\unicode[STIX]{x1D70E}}f$ of functions $f$ in Musielak–Orlicz–Morrey spaces $L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705}}(X)$ . As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

Continuity of generalized Riesz potentials for double phase functionals with variable exponents

2021 ◽  
Vol 56 (2) ◽  
pp. 329-341
Author(s):  
Takao Ohno ◽  
◽  
Tetsu Shimomura ◽  
Keyword(s):  
Morrey Spaces ◽  
Variable Exponents ◽  
Riesz Potentials ◽  
Double Phase

In this note, we discuss the continuity of generalized Riesz potentials \( I_{\rho}f\) of functions in Morrey spaces \(L^{\Phi,\nu(\cdot)}(G)\) of double phase functionals with variable exponents.

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