Lavrentiev gap for some classes of generalized Orlicz functions

Crouzeix-Raviart finite element method for non-autonomous variational problems with Lavrentiev gap

“…Since the Newton scheme struggles with the computation of the discrete minimiser, we utilise a fixed point iteration similar to the one introduced in [Die+20] without regularisation. The paper [BS21] proves that the W -minimum grows asymptotically slightly slower than the H-minimum with respect to the scaling of the boundary data ψ = λu 0 , leading to a gap for all sufficiently large parameters λ. Our numerical experiments, displayed in Figure 3, indicate a gap for all λ > 0.…”

“…(4) Borderline case of double phase potential (Balci-Surnachev [BS21]). This example involves constants β > 1, γ > 1 and the weight function a from the previous example with α = 0.…”

Crouzeix-Raviart finite element method for non-autonomous variational problems with Lavrentiev gap

“…(A1)-(A3) or (B) on p. 4). These assumptions are valid for a wide class of variational problems that exhibit a Lavrentiev gap, including important models such as the double-phase potential [5,10,13,21,24,42], the variable exponent Laplacian [27,41,42], and the weighted p-energy [28]. The problems cited above do not exhaust the full range of present research in numerical analysis for problems with non-standard growth (for example, a posteriori estimates for variational problems with nonstandard power functionals [36]).…”

Continuity at a boundary point of solutions to quasilinear elliptic equations with generalized Orlicz growth and non-logarithmic conditions

“…4) degenerate double phase g(x, v) = v p−1 1+ b(x) ln(1+ v) , 0 b(x) ∈ L ∞ (Ω), cf. [5,7,12], 5) double variable exponent g(x, v) = v p(x)−1 + v q(x)−1 , cf. [13,52,60],…”

“…Recently, many studies have been devoted to questions of regularity under Zhikov's [61] and Fan's [20] logarithmic condition, when λ(r) L < +∞ in (1.5)-(1.7) (see e.g. [1,[5][6][7][8]12,15,16,21,23,35,46,47,50,57]), and now the elliptic theory has reached a completely satisfactory form (see original surveys [29,41,42,48] and monograph [49]). In general, the logarithmic condition has significantly advanced the theory of function spaces with variable exponents [17,19], which are an integral part of the generalized Orlicz (Musielak-Orlicz) spaces [27,45].…”