Regularity results for a priori bounded minimizers of non-autonomous functionals with discontinuous coefficients

2019 ◽  
Vol 12 (1) ◽  
pp. 85-110 ◽  
Author(s):  
Raffaella Giova ◽  
Antonia Passarelli di Napoli
Keyword(s):  
Sobolev Space ◽  
A Priori ◽  
Elliptic Systems ◽  
Discontinuous Coefficients ◽  
Growth Conditions ◽  
Integral Functionals ◽  
Local Minimizers ◽  
Higher Integrability ◽  
Regularity Results ◽  
Higher Differentiability

AbstractWe prove the higher differentiability and the higher integrability of the a priori bounded local minimizers of integral functionals of the form\mathcal{F}(v,\Omega)=\int_{\Omega}f(x,Dv(x))\,{\mathrm{d}}x,with convex integrand satisfyingp-growth conditions with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to thex-variable belongs to a suitable Sobolev space. The a priori boundedness of the minimizers allows us to obtain the higher differentiability under a Sobolev assumption which is independent on the dimensionnand that, in the case{p\leq n-2}, improves previous known results. We also deal with solutions of elliptic systems with discontinuous coefficients under the so-called Uhlenbeck structure. In this case, it is well known that the solutions are locally bounded and therefore we obtain analogous regularity results without the a priori boundedness assumption.

scholarly journals Existence and regularity results for weak solutions to (p,q)-elliptic systems in divergence form

2018 ◽  
Vol 11 (3) ◽  
pp. 273-288 ◽  
Author(s):  
Miroslav Bulíček ◽  
Giovanni Cupini ◽  
Bianca Stroffolini ◽  
Anna Verde
Keyword(s):  
Weak Solutions ◽  
Elliptic Systems ◽  
Growth Conditions ◽  
Divergence Form ◽  
Variational Structure ◽  
Non Linear ◽  
Existence And Regularity Results ◽  
Regularity Results ◽  
Higher Differentiability

AbstractWe prove existence and regularity results for weak solutions of non-linear elliptic systems with non-variational structure satisfying {(p,q)}-growth conditions. In particular, we are able to prove higher differentiability results under a dimension-free gap between p and q.

scholarly journals Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrea Gentile
Keyword(s):  
Sobolev Space ◽  
Regularity Result ◽  
Growth Conditions ◽  
Quadratic Growth ◽  
Lipschitz Regularity ◽  
Higher Integrability ◽  
Higher Differentiability

AbstractWe consider functionals of the form\mathcal{F}(v,\Omega)=\int_{\Omega}f(x,Dv(x))\,dx,with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space {W^{1,q}}. We prove a higher differentiability result for the minimizers. We also infer a Lipschitz regularity result of minimizers if {q>n}, and a result of higher integrability for the gradient if {q=n}. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

scholarly journals Monotonicity and rigidity of solutions to some elliptic systems with uniform limits

2019 ◽  
Vol 22 (05) ◽  
pp. 1950044 ◽  
Author(s):  
Alberto Farina ◽  
Berardino Sciunzi ◽  
Nicola Soave
Keyword(s):  
A Priori ◽  
Elliptic Systems ◽  
A Priori Bounds ◽  
Liouville Type ◽  
Liouville Type Theorems ◽  
Regularity Results

In this paper, we prove the validity of Gibbons’ conjecture for a coupled competing Gross–Pitaevskii system. We also provide sharp a priori bounds, regularity results and additional Liouville-type theorems.

scholarly journals Singular quasilinear convective elliptic systems in ℝ N

2022 ◽  
Vol 11 (1) ◽  
pp. 741-756
Author(s):  
Umberto Guarnotta ◽  
Salvatore Angelo Marano ◽  
Abdelkrim Moussaoui
Keyword(s):  
Fixed Point ◽  
Weak Solution ◽  
Elliptic System ◽  
A Priori Estimates ◽  
A Priori ◽  
Elliptic Systems ◽  
Perturbation Techniques ◽  
Priori Estimates ◽  
Regularity Results

Abstract The existence of a positive entire weak solution to a singular quasi-linear elliptic system with convection terms is established, chiefly through perturbation techniques, fixed point arguments, and a priori estimates. Some regularity results are then employed to show that the obtained solution is actually strong.

scholarly journals Higher integrability for variational integrals with non-standard growth

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Mathias Schäffner
Keyword(s):  
Partial Regularity ◽  
Growth Conditions ◽  
Integral Functionals ◽  
Restrictive Assumption ◽  
Higher Integrability ◽  
Variational Integrals

AbstractWe consider autonomous integral functionals of the form $$\begin{aligned} {\mathcal {F}}[u]:=\int _\varOmega f(D u)\,dx \quad \text{ where } u:\varOmega \rightarrow {\mathbb {R}}^N, N\ge 1, \end{aligned}$$ F [ u ] : = ∫ Ω f ( D u ) d x where u : Ω → R N , N ≥ 1 , where the convex integrand f satisfies controlled (p, q)-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of $${\mathcal {F}}$$ F assuming $$\frac{q}{p}<1+\frac{2}{n-1}$$ q p < 1 + 2 n - 1 , $$n\ge 3$$ n ≥ 3 . This improves earlier results valid under the more restrictive assumption $$\frac{q}{p}<1+\frac{2}{n}$$ q p < 1 + 2 n .

scholarly journals Growth conditions and regularity, an optimal local boundedness result

2020 ◽  
pp. 2050029
Author(s):  
Jonas Hirsch ◽  
Mathias Schäffner
Keyword(s):  
Growth Conditions ◽  
Integral Functionals ◽  
Local Minimizers ◽  
Local Boundedness ◽  
Scalar Integral ◽  
Boundedness Result

We prove local boundedness of local minimizers of scalar integral functionals [Formula: see text], [Formula: see text] where the integrand satisfies [Formula: see text]-growth of the form [Formula: see text] under the optimal relation [Formula: see text].

On higher differentiability of solutions of parabolic systems with discontinuous coefficients and (p, q)-growth

2019 ◽  
Vol 150 (1) ◽  
pp. 419-451
Author(s):  
Flavia Giannetti ◽  
Antonia Passarelli di Napoli ◽  
Christoph Scheven
Keyword(s):  
Weak Solutions ◽  
A Priori ◽  
A Priori Estimate ◽  
Parabolic Systems ◽  
Discontinuous Coefficients ◽  
Sobolev Type ◽  
Dirichlet Problems ◽  
Priori Estimate ◽  
Image Position ◽  
Higher Differentiability

AbstractWe consider weak solutions $u:\Omega _T\to {\open R}^N$ to parabolic systems of the type $$u_t-{\rm div}\;a(x,t,Du) = 0\quad {\rm in}\;\Omega _T = \Omega \times (0,T),$$where the function a(x, t, ξ) satisfies (p, q)-growth conditions. We give an a priori estimate for weak solutions in the case of possibly discontinuous coefficients. More precisely, the partial maps $x\mapsto a(x,t,\xi )$ under consideration may not be continuous, but may only possess a Sobolev-type regularity. In a certain sense, our assumption means that the weak derivatives $D_xa(\cdot ,\cdot ,\xi )$ are contained in the class $L^\alpha (0,T;L^\beta (\Omega ))$, where the integrability exponents $\alpha ,\beta $ are coupled by $$\displaystyle{{p(n + 2)-2n} \over {2\alpha }} + \displaystyle{n \over \beta } = 1-\kappa $$for some κ ∈ (0,1). For the gap between the two growth exponents we assume $$2 \les p < q \les p + \displaystyle{{2\kappa } \over {n + 2}}.$$Under further assumptions on the integrability of the spatial gradient, we prove a result on higher differentiability in space as well as the existence of a weak time derivative $u_t\in L^{p/(q-1)}_{{\rm loc}}(\Omega _T)$. We use the corresponding a priori estimate to deduce the existence of solutions of Cauchy–Dirichlet problems with the mentioned higher differentiability property.

scholarly journals Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrea Gentile
Keyword(s):  
Besov Space ◽  
Lipschitz Space ◽  
Growth Conditions ◽  
Obstacle Problems ◽  
Quadratic Growth ◽  
Regularity Assumption ◽  
Lipschitz Regularity ◽  
Fixed Boundary ◽  
Higher Differentiability ◽  
Admissible Functions

Abstract We establish some higher differentiability results of integer and fractional order for solutions to non-autonomous obstacle problems of the form min ⁡ { ∫ Ω f ⁢ ( x , D ⁢ v ⁢ ( x ) ) : v ∈ K ψ ⁢ ( Ω ) } , \min\biggl{\{}\int_{\Omega}f(x,Dv(x)):v\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where the function 𝑓 satisfies 𝑝-growth conditions with respect to the gradient variable, for 1 < p < 2 1<p<2 , and K ψ ⁢ ( Ω ) \mathcal{K}_{\psi}(\Omega) is the class of admissible functions v ∈ u 0 + W 0 1 , p ⁢ ( Ω ) v\in u_{0}+W^{1,p}_{0}(\Omega) such that v ≥ ψ v\geq\psi a.e. in Ω, where u 0 ∈ W 1 , p ⁢ ( Ω ) u_{0}\in W^{1,p}(\Omega) is a fixed boundary datum. Here we show that a Sobolev or Besov–Lipschitz regularity assumption on the gradient of the obstacle 𝜓 transfers to the gradient of the solution, provided the partial map x ↦ D ξ ⁢ f ⁢ ( x , ξ ) x\mapsto D_{\xi}f(x,\xi) belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with sub-quadratic growth conditions with respect to the gradient variable, i.e. f ⁢ ( x , ξ ) ≈ a ⁢ ( x ) ⁢ | ξ | p f(x,\xi)\approx a(x)\lvert\xi\rvert^{p} with 1 < p < 2 1<p<2 , and where the map 𝑎 belongs to a Sobolev or Besov–Lipschitz space.

Sign in / Sign up

Export Citation Format

Share Document