We prove the existence of a solution to a quasilinear system of degenerate equations, when the datum is in a Marcinkiewicz space. The main assumption asks the off-diagonal coefficients to have support in the union of a geometric progression of squares.
Citation: Patrizia Di Gironimo, Salvatore Leonardi, Francesco Leonetti, Marta Macrì, Pier Vincenzo Petricca. Existence of solutions to some quasilinear degenerate elliptic systems with right hand side in a Marcinkiewicz space[J]. Mathematics in Engineering, 2023, 5(3): 1-23. doi: 10.3934/mine.2023055
[1] | Li Shen, Jian Liu, Guo-Wei Wei . Evolutionary Khovanov homology. AIMS Mathematics, 2024, 9(9): 26139-26165. doi: 10.3934/math.20241277 |
[2] | Linlin Tan, Meiying Cui, Bianru Cheng . An approach to the global well-posedness of a coupled 3-dimensional Navier-Stokes-Darcy model with Beavers-Joseph-Saffman-Jones interface boundary condition. AIMS Mathematics, 2024, 9(3): 6993-7016. doi: 10.3934/math.2024341 |
[3] | Mei Li, Wanqiang Shen . Integral method from even to odd order for trigonometric B-spline basis. AIMS Mathematics, 2024, 9(12): 36470-36492. doi: 10.3934/math.20241729 |
[4] | Oussama Bouanani, Salim Bouzebda . Limit theorems for local polynomial estimation of regression for functional dependent data. AIMS Mathematics, 2024, 9(9): 23651-23691. doi: 10.3934/math.20241150 |
[5] | Rolly Czar Joseph Castillo, Renier Mendoza . On smoothing of data using Sobolev polynomials. AIMS Mathematics, 2022, 7(10): 19202-19220. doi: 10.3934/math.20221054 |
[6] | Yunan He, Jian Liu . Multi-scale Hochschild spectral analysis on graph data. AIMS Mathematics, 2025, 10(1): 1384-1406. doi: 10.3934/math.2025064 |
[7] | Wan Anis Farhah Wan Amir, Md Yushalify Misro, Mohd Hafiz Mohd . Flexible functional data smoothing and optimization using beta spline. AIMS Mathematics, 2024, 9(9): 23158-23181. doi: 10.3934/math.20241126 |
[8] | Zhe Su, Yiying Tong, Guo-Wei Wei . Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning. AIMS Mathematics, 2024, 9(10): 27438-27470. doi: 10.3934/math.20241333 |
[9] | Raju Doley, Saifur Rahman, Gayatri Das . On knot separability of hypergraphs and its application towards infectious disease management. AIMS Mathematics, 2023, 8(4): 9982-10000. doi: 10.3934/math.2023505 |
[10] | Chao Wang, Fajie Wang, Yanpeng Gong . Analysis of 2D heat conduction in nonlinear functionally graded materials using a local semi-analytical meshless method. AIMS Mathematics, 2021, 6(11): 12599-12618. doi: 10.3934/math.2021726 |
We prove the existence of a solution to a quasilinear system of degenerate equations, when the datum is in a Marcinkiewicz space. The main assumption asks the off-diagonal coefficients to have support in the union of a geometric progression of squares.
Abbreviations
MYA: Million years ago
MSY: Male specific region of the Y
XCR: X conserved region
MSCI: Meiotic sex chromosome inactivation
XCI: X-chromosome inactivation
Xp: Paternally inherited X chromosome
Xi: Inactive X chromosome
XIC: X Inactivation Center
RepA: Repeat A
Rsx: RNA on the silent X
The class Mammalia (mammals) is divided in to two subclasses: Prototheria (monotremes) and Theria (marsupials and eutherians). All therian mammals have male heterogamety, with an XX female: XY male sex chromosome system (Figure 1), or some simple variant. This sex chromosome system arose before the marsupial/eutherian split, ~ 180 million years ago (MYA) [1], from an ordinary pair of autosomes after a mutation in the Sox3 gene resulted in the birth of the testis-determining gene Sry [2].
Genes advantageous to males accumulated on the proto Y near Sry either by transposition from autosomal sites or by mutation of existing genes. Recombination with the X was suppressed across this region so that the male advantageous genes were only inherited with the testis-determining gene, giving rise to the male specific region of the Y (MSY). Lack of recombination led to progressive gene loss on, and degradation of, the MSY [3]. Thus, in all therian species the X and Y are morphologically distinct.
The marsupial X chromosome is ~ 2/3 the size of the eutherian X, and is homologous to the long arm (Xq) and proximal short arm (Xp) of human X chromosome (called the X conserved region; XCR). In contrast, the short arm of the human X chromosome (distal to Xp11.22) is orthologues to marsupial autosomes, so was added to the eutherian X before the radiation of eutherians (~ 105 MYA), but after their divergence from marsupials [4] (Figure 1).
Monotremes, which diverged from therian mammals ~ 200 MYA, comprise one extant platypus and four extant echidna species, all with a complex of serially translocated sex chromosomes. In the model monotreme, platypus (Ornithorhychus anatinus), males have five X and five Y chromosomes, and females have 5 pairs of X chromosomes [5]. These sex chromosomes do not bear the Sry gene or share homology with the sex chromosome of therian mammals [6] (Figure 1). Instead, they share extensive homology with the independently evolved bird ZW sex chromosome system [6, 7]. Thus, sex chromosomes have evolved multiple times throughout amniote evolution [1] (Figure 1).
In spite of the lethal effect whole chromosome monosomy has for any autosome [8], such grand sex chromosome imbalances are present in many distantly related species. Ohno [9]suggested that copy number imbalance of the X with the autosomes (1X: 2A) in males resulted in the almost twofold upregulation of the X. This led to overexpression from the two Xs in females, which resulted in down-regulation of one X in that sex [10].
Upregulation of expression from the single X in male is observed in marsupials, where average transcriptional output is near diploid expression levels [11]. However, whether or not the single X is upregulated in male eutherian mammals has remained controversial as a result of inconsistent processing, filtering and analysis methods of transcriptome data [12]. The debate surrounding Ohno’s hypothesis [13, 14, 15, 16, 17, 18] has spawned a novel view that eutherian dosage compensation evolved to restore balance of X genes, which function in protein complexes or protein networks, with their autosomal partners [12]. In some instances expression of X genes were increased to match the original autosomal level (as proposed by Ohno), and in other instances expression of the autosomal genes was decreased to match the new reduced X level This suggests that hyper-expression evolved on a gene-by-gene basis and affected only a subset of X genes.
In female eutherians and marsupials, down-regulation of X genes to restore parity with the autosomes is achieved by X-chromosome inactivation (XCI); an epigenetic mechanism by which one of the two X chromosomes is silenced in somatic cells. Once silencing has occurred, it is stably maintained throughout all ensuing cell divisions [19]. Although some features that characterize the inactive X (Xi) chromosome are shared between the two lineages, lineage-specific genetic, and epigenetic differences exist [20, 21, 22, 23, 24, 25]. These similarities and differences provide insight into the evolution of mammal XCI.
In the male germline of both eutherians and marsupials, sex chromosomes are inactivated during meiosis in a process called meiotic sex chromosome inactivation (MSCI) [26, 27, 28, 29, 30]. All X-borne genes tested in opossum round spermatids were reactivated and expressed [28]. In mouse reactivation is only observed for some genes after spermatogenesis, so the paternally inherited X chromosome (Xp) is delivered to the ovum in a partially pre-inactivated state [31, 32]. After fertilization, transcription of repetitive elements on the Xp is suppressed [33], but biallelic expression is observed for X-borne genes at the two-cell stage [33, 34, 35, 36].
During eutherian mammal XCI, the choice of which X is to be inactivated can be either random with regard to the parent of origin, or imprinted, where the paternal X is inactivated in all cells. These two forms of XCI are species specific, but can also occur in different cell types within the same species. During mouse pre-implantation development, exclusive silencing of the Xp leads to establishment of imprinted-XCI [37, 38, 39]through to the blastocyst stage. Beyond this, imprinted XCI is maintained only in the trophectodermal extra-embryonic cell lineages that give rise to placental tissue, and the primitive endoderm that gives rise to the visceral endoderm and yolk sac [40]. In contrast, in the developing inner cell mass, which gives rise to the embryo proper, the inactive Xp is reactivated, which is then followed by random XCI [40, 41, 42]. Imprinted XCI was also observed in extra-embryonic cell lineages of rat [43] and cow [44, 45]. However, in human, monkey, horse, pig, mule and rabbit random XCI was observed in both embryonic and extra-embryonic cells [46, 47].
One of the marked differences between marsupial and eutherian XCI is the choice of X to be inactivated in the embryo proper. XCI in marsupial extra-embryonic, fetal, and adult tissues is imprinted, with the paternally derived X always silenced [24, 25, 48]. The reason and cause for this difference in choice during X inactivation is not understood.
Although the marsupial inactive X shares some similarities with the eutherian Xi at the cytogenetic level, such as late replication at S phase and heterochromatinization [49, 50, 51, 52, 53, 54], it differs at the molecular level [22, 25]. There are considerable differences in the histone profile of the inactive X between eutherians and marsupials [21] (Table 1), but in general at the onset of XCI, the inactive X loses epigenetic modifications associated with active transcription (e.g. H3K9ac, H4Kac and H3K4me2) and sequentially acquires repressive marks characteristics of silenced chromatin (e.g. H4K20me1 and H3K27me3). In addition, the eutherian Xi exhibits enrichment of histone variants such as macro-H2A, and hypermethylation of promoter sequences stabilizes inactivation once repression has occurred [55] (Table 1). The Xi in marsupial female possum and potoroo metaphase appears hypomethylated [23], in addition promoter DNA methylation appears absent on the Xi for loci tested in opossum [25, 56].
![]() |
The final outcome of these modifications is silencing of transcription from most genes on the Xi. However, some genes escape inactivation, and as a result are expressed from both active and inactive X chromosomes [57]. The chromatin state of these genes more closely resembles that of expressed genes on the active X and autosomes, than that of silent Xi loci. The number and identity of genes that escape inactivation is different between species. In human somatic cells 15% of genes on the X escape inactivation [58, 59, 60], with a higher frequency on the short arm (orthologues to marsupial autosomes) than on the long arm of the human X (homologous to the marsupial X). Furthermore, about 10% of human X-borne genes have variable inactivation status between tissues and/or individuals [60, 61]. In mouse somatic cells, almost all X-borne genes are inactivated; only 3% escape [59, 62].
In female mouse trophoblast stem cells [63] and extra-embryonic endoderm [64], both of which are subject to imprinted XCI, a larger number of X-borne genes (13% and 15%) are expressed from both X chromosomes. However, different subsets of genes in these extra-embryonic cell lineages are subject to XCI. The inactive Xp in mouse extra-embryonic tissues globally accumulates the same repressive histone marks as the Xi in other somatic cell types [36, 64, 65]. However, the order in which these modifications appear on the Xi is different. In random XCI enrichment of macro-H2A is a late stage event. In contrast, during imprinted XCI enrichment of H3K27me3 and macro-H2A appear early on the Xi, whereas H3K9me2 accumulation is detected later [42, 66]. Similar to Xi in somatic cells, X-borne promoters are hypermethylated on the inactive X chromosome [67].
In marsupials, a large proportion of X-borne genes escape XCI [20, 25], as in the mouse extra-embryonic membranes. Genes on the marsupial Xi exhibit variable levels of incomplete silencing across species, tissue and developmental stage [68, 69]. Approximately 15% of genes on the American grey short-tailed opossum X escape inactivation [25]. As such imprinted XCI in marsupials and mouse extra-embryonic tissues is not as complete as random XCI in eutherian somatic tissue. Interestingly, the marsupial Xi lacks the repressive H4K20me1 mark [21], which accumulates on Xi during both imprinted and random XCI in mouse [70] (Table 1). Localization of macro-H2A to the marsupial Xi is unknown.
The silencing achieved during XCI is triggered by long-noncoding (lnc) RNAs that interact with chromatin regulatory complexes to alter chromosome conformation. Yet despite the central role of RNA-chromatin interactions during XCI, they are not fully understood.
A region on the eutherian X chromosome called the X Inactivation Centre (XIC) is of key importance in coordinating XCI. The XIC contains several pseudogenes (e.g. Fxyd6) and protein-coding genes (e.g. CDX4, CHIC1, SLC16A2) [71], along with the key non-coding RNA genes (e.g. XIST, TSIX, FTX, JPX and others) (Figure 2). The XIC lncRNAs are poorly conserved between eutherian species, with the master regulator (XIST) the most conserved element between sequenced eutherian genomes [72, 73, 74]. However, there is no ortholog of XIST in the marsupial or monotreme genomes, in which the XIC locus has been disrupted [72, 75]. Interestingly, in chicken the locus remains intact with protein coding genes that share homology to XIST and the mouse Tsx gene [73].
Several exons of the chicken protein-coding gene Lnx3 share homology with the XIST gene [73, 76, 77] (Figure 3). These homologies reveal that the XIST promoter evolved from exons 1 and 2 of the Lnx3 gene, which is among the most conserved regions of the XIST gene between different eutherian species [73, 76, 77]. The remaining XIST exons (that share no homology with Lnx3) are likely to have originated via transposition of various mobile elements, presumably endogenous retroviruses, fragments of which were amplified to generate several simple tandem repeats [76, 78]. The lack of XIST in marsupials, along with it being in all eutherian genomes, means that XIST evolved as a key player in XCI in the eutherian ancestor.
The ancestral XIST gene presumably consisted of ten exons [74, 76]. Two large exons (1 and 8) together constitute about 90% of XIST and contain tandem repeats (denoted A to H) [74] (Figure 3). Tandem repeat A is conserved in all eutherian species [75, 76, 78, 79, 80, 81, 82, 83, 84], whereas presence, absence, or amplification of the other repeats is species specific [76, 83]. Six of the ancestral exons are conserved across Eutheria [73, 76, 77, 78], with the remaining four (2, 6, 9, 10) either functional- or pseudo-exons depending on species [74, 85]. Thus, human XIST encodes a 19kb transcript [86], whereas the mouse Xist transcript is 17 kb [87]. Despite variable intron-exon structures between species, XIST exons are GC dinucleotide rich compared to the introns. The proportion of GC richness is constant between species (39.8-42.2%) and similar to the whole XIC locus.
In mouse, Xist is transcribed only in females by RNA polymerase II, solely from the Xi. Analysis of the CpG dinucleotide methylation patterns in the promoter region has shown that the active Xist allele (on the inactive X) is completely unmethylated [88]. In contrast, the silent maternal Xist allele is fully methylated [88]. Although Xist RNA is spliced and polyadenylated, it is absent from polysomes [80, 89] and remains in the nucleus where it coats and forms a “Xist cloud” on the X to be inactivated [80, 90], the spreading of Xist RNA along one X chromosome in cis initiates the chromosome-wide silencing.
LINE1 retrotransposons are enriched on the X (mouse X ~ 28.5%, autosomes ~ 14.6%) [91], so were proposed to be anchor points for Xist to ensure efficient spreading of the machinery responsible for silencing [92]. A significant decrease in LINE1 density at regions containing genes that escape inactivation [93]supports this hypothesis, although a less direct role for LINEs in the spreading process seems more probable [94]. Accordingly, LINEs were proposed to moderate spatial organization of the transcriptionally silent nuclear territory of the inactive X chromosome, into which X-borne genes are recruited as they are silenced [95, 96].
Xistexpression is followed by the formation of a repressive chromatin state that excludes transcriptional machinery from the inactive X [95]. Repeat A (RepA), at the 5' end of Xist, recruits the polycomb repressive complexes PRC1 and PRC2 to the Xi. Polycomb repressive complexes decorate the Xi and catalyze the characteristic repressive histone modifications of Xi. A number of other proteins are also localized to the Xi, potentially trafficked via Xist RNA, including nuclear scaffolding factors such as SAF-A [97] and the histone variant macro-H2A [98, 99, 100, 101].
During the morula/blastocyst stage in mice, a few days after initiation of imprinted XCI, Tsix is expressed exclusively from the maternally derived X chromosome to inhibit expression from the maternally derived Xist [102, 103, 104]. However, during random XCI Tsix demarcates whichever X remains active (Xa) [105, 106, 107] and its expression prevents in cis transcription of Xist and, ultimately, inactivation of that X [104]. Tsix in rodents spans more than a 40 kb region that encompasses the entire transcription unit of Xist [108]. In primates, cow and dog there are many species-specific repeat-element insertions, and large deletions, that disrupt the overall structure of TSIX [109]. In human, TSIX appear to be an expressed pseudogene unable to repress XIST, and overlaps only with the last two exons of XIST [109, 110]. In contrast to mouse Tsix, the human TSIX is expressed with XIST from the Xi in the fetal cells, throughout gestation, but cease transcription shortly after birth [110]. Thus, Tsix function seems limited to rodents.
Ftx and Jpx potentially upregulate Xist, and are conserved in mouse, human, and cow [71, 111], evolving from the protein coding genes Wave4 and Uspl, respectively [76]. Both genes escape imprinted XCI and are expressed predominantly from the paternal allele at the pre-implantation stage [112]. Deletion of the Ftx promoter leads to decreased Xist expression in male embryonic stem (ES) cells [111],indicating that it is a positive regulator of Xist. However a recent study shows that Ftx disruption did not affect embryo survival, or expression of Xist and other X-borne genes during pre-implantation, thus is dispensable for imprinted inactivation [113]. Whether Ftx is involved in random XCI in post-implantation embryos is yet to be determined.
Jpx is located just downstream of the Ftx locus, and approximately 10 kb upstream of Xist [71, 114, 115]. Jpx escapes XCI and can upregulate Xist expression on the Xi [71, 115, 116] by evicting CTCF from the Xist promoter [117]. Deletion of a single Jpx allele in XX female ES cells results in failed accumulation of Xist on either X, and inactivation is prevented [116].
Although the function of XIST may be well conserved in eutherians, other elements of the XIC (even those with sequence conservation) may not have conserved function (Table 2). However, poor sequence conservation of noncoding RNAs in the XIC does not necessarily indicate a lack of function [118, 119], as maintaining secondary structure (and therefore function) of lncRNA molecules may only require short stretches of sequence preservation. This poor conservation might indicate their adaptation to function in specific genomic environments, suggesting that regulation of XCI is at least partially species-specific. For instance, the recently evolved lncRNA gene XACT is only present in human and chimpanzee, but not in macaques or more distantly related species [120]. XACT is expressed from and coats the active X in female human embryonic stem cells and early differentiating cells, and may contribute to protecting the Xa from inactivation [120].
Eutherian | Marsupial | |||
Mouse | Human | Opossum | ||
lncRNAs | Xist | √ | √ | × |
Ftx | √ | √ | × | |
Jpx | √ | √ | × | |
Tsix | √ | * | × | |
XACT | × | √ | × | |
Rsx | × | × | √ | |
√ = presence, × = absence, * = pseudogene. |
The marsupial Lnx3 (the precursor of XIST) gene has a native open reading frame that is expressed in both males and females. The eutherian XIC locus is disrupted in marsupials, and Lnx3 presumably functions as a PDZ domain containing ring finger protein rather than an untranslated nuclear RNA similar to XIST. Consequently, the X inactivation process in marsupials involves neither XIST nor the XIC.
The marsupial X chromosome has multiple large-scale internal rearrangements with respect to both the human X, and between Australian and American representative species [121]. This is contrary to the generally conserved gene order on the eutherian X [122, 123], presumably due to purifying selection against rearrangements that perturb interactions between XIST and regions of the X intended for inactivation [93]. Extensive rearrangement of the marsupial X chromosome was taken as support for the lack of a XIST equivalent [121]. However, Rsx (RNA on the silent X) was identified in opossum (and two Australian marsupials), and appears to fulfill some of the functions of XIST [22]. As such, the epigenetic mechanisms that silence the inactive X in the somatic cells of marsupials and eutherians share a rema rkable degree of convergence.
The mature Rsx in opossum is a 27-kb non-coding RNA with several XIST-like characteristics, such as a high GC content and enrichment of conserved 5’ tandem repeats that may be involved in the formation of stem-loop structures [22]. These are potentially important functional domains necessary for directing protein complexes responsible for chromatin modification that repress transcription. However, further studies are needed to determine the candidate functional domains of Rsx.
Rsx is located adjacent to Hprt on the marsupial X in a different genomic context to—and shares no sequence homology with—XIST. Yet, like XIST, Rsx is expressed exclusively in female somatic cells [22] and extra embryonic membranes [25], but not in germ cells where both X chromosomes are active [22]. Rsx is expressed in cis from Xi, around which it forms a “Rsx-cloud” that results in repressed gene activity. Moreover, after introduction of Rsx into mouse ES cells, Rsx RNA coated the transgenic chromosome and resulted in its inactivation in more than half the cells examined [22].
Monoallelic expression from the paternally derived allele of Rsx, in both fetal brain and extra embryonic membranes, was shown to be due to different epigenetic characteristics of the active and inactive alleles. Rsx, similar to Xist, is differentially methylated at its promoter. There is ~ 100% methylation of the maternally derived allele, and virtually no methylation of the paternally derived allele [25]. Furthermore, H3K27me3 repressive mark was absent from the Rsx gene body, demonstrating that similar epigenetic mechanisms regulate the independently evolved Rsx and XIST genes [25].
Although independently evolved, there appear to be remarkable functional similarities shared by Xist and Rsx. However, it remains unknown if Rsx can perform all functions attributed to Xist, or if it traffics the epigenetic machinery as Xist is proposed to do. Since overlapping, but different, suites of repressive chromatin modifications are used to silence the X in eutherians and marsupials, many of these epigenetic tools were likely utilized in the therian ancestor to achieve X chromosome inactivation. However, it is yet to be determined if Rsx was an XCI switch in the therian mammal ancestor that was retained in marsupials, and then replaced by Xist in the eutherian ancestor; or if they evolved simultaneously in the two lineages. Perhaps the epigenetic differences observed between eutherian and marsupial XCI merely reflect that these two lncRNAs direct protein complexes that are responsible for different chromatin modification. Finally, the potential existence of a marsupial X inactivation center close to Rsx, which bears lncRNAs that may regulate Rsx expression, remains a fascinating possibility.
P.D.W. was supported by an ARC fellowship.
The authors declare no conflicts of interest in this paper.
[1] | S. Aouaoui, Solutions to quasilinear equations of N-biharmonic type with degenerate coercivity, Electronic Journal of Differential Equations, 2014 (2014), 228. |
[2] |
A. Alvino, L. Boccardo, V. Ferone, L. Orsina, G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Annali di Matematica, 182 (2003), 53–79. https://doi.org/10.1007/s10231-002-0056-y doi: 10.1007/s10231-002-0056-y
![]() |
[3] | A. Alvino, V. Ferone, G. Trombetti, A priori estimates for a class of non uniformly elliptic equations, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 381–391. |
[4] | H. Ayadi, F. Mokhtari, Nonlinear anisotropic elliptic equations with variable exponents and degenerate coercivity, Electronic Journal of Differential Equations, 2018 (2018), 45. |
[5] |
P. Baroni, M. Colombo, G. Mingione, Regularity for general functionals with double phase, Calc. Var., 57 (2018), 62. https://doi.org/10.1007/s00526-018-1332-z doi: 10.1007/s00526-018-1332-z
![]() |
[6] |
L. Beck, G. Mingione, Lipschitz bounds and nonuniform ellipticity, Commun. Pure Appl. Math., 73 (2020), 944–1034, https://doi.org/10.1002/cpa.21880 doi: 10.1002/cpa.21880
![]() |
[7] | L. Boccardo, Quasilinear elliptic equations with natural growth terms: the regularizing effect of the lower order terms, J. Nonlinear Convex Anal., 7 (2006), 355–365. |
[8] |
L. Boccardo, Marcinkiewicz estimates for solutions of some elliptic problems with nonregular data, Annali di Matematica, 188 (2009), 591–601. https://doi.org/10.1007/s10231-008-0090-5 doi: 10.1007/s10231-008-0090-5
![]() |
[9] | L. Boccardo, H. Brézis, Some remarks on a class of elliptic equations with degenerate coercivity, Boll. Unione Mat. Ital., 6 (2003), 521–530. |
[10] | L. Boccardo, G. Croce, Elliptic partial differential equations: Existence and regularity of distributional solutions, Berlin: De Gruyter, 2013. https://doi.org/10.1515/9783110315424 |
[11] |
L. Boccardo, G. Croce, L. Orsina, Existence of solutions for some noncoercive elliptic problems involving derivatives of nonlinear terms, Differential Equations & Applications, 4 (2012), 3–9. https://doi.org/10.7153/dea-04-02 doi: 10.7153/dea-04-02
![]() |
[12] |
L. Boccardo, G. Croce, L. Orsina, Nonlinear degenerate elliptic problems with W1,10(Ω) solutions, Manuscripta Math., 137 (2012), 419–439. https://doi.org/10.1007/s00229-011-0473-6 doi: 10.1007/s00229-011-0473-6
![]() |
[13] | L. Boccardo, G. Croce, C. Tanteri, An elliptic system with degenerate coercivity, Rend. Mat. Appl. (7), 36 (2015), 1–9. |
[14] | L. Boccardo, A. Dall'Aglio, L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 51–81. |
[15] |
P. Cianci, G. R. Cirmi, S. D'Asero, S. Leonardi, Morrey estimates for solutions of singular quadratic nonlinear equations, Annali di Matematica, 196 (2017), 1739–1758. https://doi.org/10.1007/s10231-017-0636-5 doi: 10.1007/s10231-017-0636-5
![]() |
[16] |
G. R. Cirmi, S. D'Asero, S. Leonardi, Fourth-order nonlinear elliptic equations with lower order term and natural growth conditions, Nonlinear Anal. Theor., 108 (2014), 66–86. https://doi.org/10.1016/j.na.2014.05.014 doi: 10.1016/j.na.2014.05.014
![]() |
[17] |
G. R. Cirmi, S. D'Asero, S. Leonardi, Morrey estimates for a class of elliptic equations with drift term, Adv. Nonlinear Anal., 9 (2020), 1333–1350. https://doi.org/10.1515/anona-2020-0055 doi: 10.1515/anona-2020-0055
![]() |
[18] |
G. R. Cirmi, S. D'Asero, S. Leonardi, On the existence of weak solutions to a class of nonlinear elliptic systems with drift term, J. Math. Anal. Appl., 491 (2020), 124370. https://doi.org/10.1016/j.jmaa.2020.124370 doi: 10.1016/j.jmaa.2020.124370
![]() |
[19] |
G. R. Cirmi, S. D'Asero, S. Leonardi, Morrey estimates for a class of noncoercive elliptic systems with VMO-coefficients, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 32 (2021), 317–334. https://doi.org/10.4171/RLM/938 doi: 10.4171/RLM/938
![]() |
[20] |
G. R. Cirmi, S. D'Asero, S. Leonardi, M. M. Porzio, Local regularity results for solutions of linear elliptic equations with drift term, Adv. Calc. Var., 15 (2022), 19–32. https://doi.org/10.1515/acv-2019-0048 doi: 10.1515/acv-2019-0048
![]() |
[21] |
G. R. Cirmi, S. Leonardi, Regularity results for the gradient of solutions linear elliptic equations with L1,λ data, Annali di Matematica, 185 (2006), 537–553. http://doi.org/10.1007/s10231-005-0167-3 doi: 10.1007/s10231-005-0167-3
![]() |
[22] |
G. R. Cirmi, S. Leonardi, Regularity results for solutions of nonlinear elliptic equations with L1,λ data, Nonlinear Anal. Theor., 69 (2008), 230–244. https://doi.org/10.1016/j.na.2007.05.014 doi: 10.1016/j.na.2007.05.014
![]() |
[23] |
G. R. Cirmi, S. Leonardi, Higher differentiability for the solutions of nonlinear elliptic systems with lower-order terms and L1,θ-data, Annali di Matematica, 193 (2014), 115–131. https://doi.org/10.1007/s10231-012-0269-7 doi: 10.1007/s10231-012-0269-7
![]() |
[24] |
G. Croce, An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term, Discrete Contin. Dyn. Syst. S, 5 (2012), 507–530. https://doi.org/10.3934/dcdss.2012.5.507 doi: 10.3934/dcdss.2012.5.507
![]() |
[25] | G. Croce, The regularizing effects of some lower order terms in an elliptic equation with degenerate coercivity, Rendiconti di Matematica, Serie Ⅶ, 27 (2007), 299–314. |
[26] |
G. Cupini, F. Giannetti, R. Giova, A. Passarelli di Napoli, Regularity results for vectorial minimizers of a class of degenerate convex integrals, J. Differ. Equations, 265 (2018), 4375–4416. https://doi.org/10.1016/j.jde.2018.06.010 doi: 10.1016/j.jde.2018.06.010
![]() |
[27] |
G. Cupini, P. Marcellini, E. Mascolo, Regularity of minimizers under limit growth conditions, Nonlinear Anal. Theor., 153 (2017), 294–310. https://doi.org/10.1016/j.na.2016.06.002 doi: 10.1016/j.na.2016.06.002
![]() |
[28] |
C. De Filippis, G. Mingione, Lipschitz bounds and nonautonomous integrals, Arch. Rational Mech. Anal., 242 (2021), 973–1057. https://doi.org/10.1007/s00205-021-01698-5 doi: 10.1007/s00205-021-01698-5
![]() |
[29] | E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital. Ser. Ⅳ, 1 (1968), 135–137. |
[30] |
F. Della Pietra, G. Di Blasio, Comparison, existence and regularity results for a class of non-uniformly elliptic equations, Differential Equations & Applications, 2 (2010), 79–103. https://doi.org/10.7153/dea-02-07 doi: 10.7153/dea-02-07
![]() |
[31] | P. Di Gironimo, F. Leonetti, M. Macrì, P. V. Petricca, Existence of bounded solutions for some quasilinear degenerate elliptic systems, Minimax Theory and its Applications, 6 (2021), 321–340. |
[32] | P. Di Gironimo, F. Leonetti, M. Macrì, P. V. Petricca, Existence of solutions to some quasilinear degenerate elliptic systems when the datum has an intermediate degree of integrability, Complex Var. Elliptic Equ., in press. https://doi.org/10.1080/17476933.2022.2069753 |
[33] |
G. Dolzmann, N. Hungerbuhler, S. Muller, Non-linear elliptic systems with measure-valued right hand side, Math. Z., 226 (1997), 545–574. https://doi.org/10.1007/PL00004354 doi: 10.1007/PL00004354
![]() |
[34] |
V. Ferone, N. Fusco, VMO solutions of the N-Laplacian with measure data, Comptes Rendus de l'Académie des Sciences Series I-Mathematics, 325 (1997), 365–370. https://doi.org/10.1016/S0764-4442(97)85618-2 doi: 10.1016/S0764-4442(97)85618-2
![]() |
[35] |
H. Gao, M. Huang, W. Ren, Regularity for entropy solutions to degenerate elliptic equations, J. Math. Anal. Appl., 491 (2020), 124251. https://doi.org/10.1016/j.jmaa.2020.124251 doi: 10.1016/j.jmaa.2020.124251
![]() |
[36] |
H. Gao, F. Leonetti, W. Ren, Regularity for anisotropic elliptic equations with degenerate coercivity, Nonlinear Anal., 187 (2019), 493–505. https://doi.org/10.1016/j.na.2019.06.017 doi: 10.1016/j.na.2019.06.017
![]() |
[37] |
D. Giachetti, M. M. Porzio, Existence results for some nonuniformly elliptic equations with irregular data, J. Math. Anal. Appl., 257 (2001), 100–130. https://doi.org/10.1006/jmaa.2000.7324 doi: 10.1006/jmaa.2000.7324
![]() |
[38] |
D. Giachetti, M. M. Porzio, Elliptic equations with degenerate coercivity: gradient regularity, Acta Math. Sinica, 19 (2003), 349–370. https://doi.org/10.1007/s10114-002-0235-1 doi: 10.1007/s10114-002-0235-1
![]() |
[39] | W. Hao, S. Leonardi, J. Nečas, An example of irregular solution to a nonlinear Euler-Lagrange elliptic system with real analytic coefficients, Annali della Scuola Normale Superiore di Pisa (Ⅳ), 23 (1996), 57–67. |
[40] | W. Hao, S. Leonardi, M. Steinhauer, Examples of discontinuous, divergence-free solutions to elliptic variational problems, Comment. Math. Univ. Carolin., 36 (1995), 511–517. |
[41] |
J. Kristensen, G. Mingione, The singular set of ω-minima, Arch. Rational Mech. Anal., 177 (2005), 93–114. https://doi.org/10.1007/s00205-005-0361-x doi: 10.1007/s00205-005-0361-x
![]() |
[42] |
J. Kristensen, G. Mingione, The singular set of minima of integral functionals, Arch. Rational Mech. Anal., 180 (2006), 331–398. https://doi.org/10.1007/s00205-005-0402-5 doi: 10.1007/s00205-005-0402-5
![]() |
[43] |
J. Kristensen, G. Mingione, Boundary regularity in variational problems, Arch. Rational Mech. Anal., 198 (2010), 369–455. https://doi.org/10.1007/s00205-010-0294-x doi: 10.1007/s00205-010-0294-x
![]() |
[44] |
T. Kuusi, G. Mingione, Universal potential estimates, J. Funct. Anal., 262 (2012), 4205–4269. https://doi.org/10.1016/j.jfa.2012.02.018 doi: 10.1016/j.jfa.2012.02.018
![]() |
[45] |
T. Kuusi, G. Mingione, Linear potentials in nonlinear potential theory, Arch. Rational Mech. Anal., 207 (2013), 215–246. https://doi.org/10.1007/s00205-012-0562-z doi: 10.1007/s00205-012-0562-z
![]() |
[46] |
T. Kuusi, G. Mingione, Guide to nonlinear potential estimates, Bull. Math. Sci., 4 (2014), 1–82. https://doi.org/10.1007/s13373-013-0048-9 doi: 10.1007/s13373-013-0048-9
![]() |
[47] |
T. Kuusi, G. Mingione, The Wolff gradient bound for degenerate parabolic equations, J. Eur. Math. Soc., 16 (2014), 835–892. https://doi.org/10.4171/JEMS/449 doi: 10.4171/JEMS/449
![]() |
[48] |
T. Kuusi, G. Mingione, Vectorial nonlinear potential theory, J. Eur. Math. Soc., 20 (2018), 929–1004. https://doi.org/10.4171/JEMS/780 doi: 10.4171/JEMS/780
![]() |
[49] |
S. Leonardi, On constants of some regularity theorems. De Giorgi's type counterexample, Math. Nachr., 192 (1998), 191–204. https://doi.org/10.1002/mana.19981920111 doi: 10.1002/mana.19981920111
![]() |
[50] |
S. Leonardi, Gradient estimates below duality exponent for a class of linear elliptic systems, Nonlinear Differ. Equ. Appl., 18 (2011), 237–254. http://doi.org/10.1007/s00030-010-0093-y doi: 10.1007/s00030-010-0093-y
![]() |
[51] |
S. Leonardi, Morrey estimates for some classes of elliptic equations with a lower order term, Nonlinear Anal., 177 (2018), 611–627. https://doi.org/10.1016/j.na.2018.05.010 doi: 10.1016/j.na.2018.05.010
![]() |
[52] |
S. Leonardi, F. Leonetti, C. Pignotti, E. Rocha, V. Staicu, Maximum principles for some quasilinear elliptic systems, Nonlinear Anal., 194 (2020), 111377. https://doi.org/10.1016/j.na.2018.11.004 doi: 10.1016/j.na.2018.11.004
![]() |
[53] | S. Leonardi, F. Leonetti, C. Pignotti, E. Rocha, V. Staicu, Local boundedness for weak solutions to some quasilinear elliptic systems, Minimax Theory and its Applications, 6 (2021), 365–378. |
[54] |
S. Leonardi, F. Leonetti, E. Rocha, V. Staicu, Butterfly support for off diagonal coefficients and boundedness of solutions to quasilinear elliptic systems, Adv. Nonlinear Anal., 11 (2022), 672–683. https://doi.org/10.1515/anona-2021-0205 doi: 10.1515/anona-2021-0205
![]() |
[55] |
F. Leonetti, P. V. Petricca, Existence of bounded solutions to some nonlinear degenerate elliptic systems, Discrete Contin. Dyn. Syst. B, 11 (2009), 191–203. https://doi.org/10.3934/dcdsb.2009.11.191 doi: 10.3934/dcdsb.2009.11.191
![]() |
[56] |
F. Leonetti, E. Rocha, V. Staicu, Quasilinear elliptic systems with measure data, Nonlinear Anal. Theor., 154 (2017), 210–224. https://doi.org/10.1016/j.na.2016.04.002 doi: 10.1016/j.na.2016.04.002
![]() |
[57] |
F. Leonetti, E. Rocha, V. Staicu, Smallness and cancellation in some elliptic systems with measure data, J. Math. Anal. Appl., 465 (2018), 885–902. https://doi.org/10.1016/j.jmaa.2018.05.047 doi: 10.1016/j.jmaa.2018.05.047
![]() |
[58] |
F. Leonetti, R. Schianchi, A remark on some degenerate elliptic problems, Ann. Univ. Ferrara, 44 (1998), 123–128. https://doi.org/10.1007/BF02828019 doi: 10.1007/BF02828019
![]() |
[59] |
J. Leray, J. L. Lions, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bulletin de la Société Mathématique de France, 93 (1965), 97–107. https://doi.org/10.24033/bsmf.1617 doi: 10.24033/bsmf.1617
![]() |
[60] |
V. G. Maz'ja, Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients, Funct. Anal. Appl., 2 (1968), 230–234. https://doi.org/10.1007/BF01076124 doi: 10.1007/BF01076124
![]() |
[61] |
G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations, Appl. Math., 51 (2006), 355–426. https://doi.org/10.1007/s10778-006-0110-3 doi: 10.1007/s10778-006-0110-3
![]() |
[62] |
G. Mingione, The Calderˊon-Zygmund theory for elliptic problems with measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 6 (2007), 195–261. https://doi.org/10.2422/2036-2145.2007.2.01 doi: 10.2422/2036-2145.2007.2.01
![]() |
[63] |
G. Mingione, Gradient estimates below the duality exponent, Math. Ann., 346 (2010), 571–627. https://doi.org/10.1007/s00208-009-0411-z doi: 10.1007/s00208-009-0411-z
![]() |
[64] |
G. Mingione, Gradient potential estimates, J. Eur. Math. Soc., 13 (2011), 459–486. https://doi.org/10.4171/JEMS/258 doi: 10.4171/JEMS/258
![]() |
[65] |
G. Mingione, G. Palatucci, Developments and perspectives in nonlinear potential theory, Nonlinear Anal., 194 (2020), 111452. https://doi.org/10.1016/j.na.2019.02.006 doi: 10.1016/j.na.2019.02.006
![]() |
[66] | J. Nečas, J. Stará, Principio di massimo per i sistemi ellittici quasi-lineari non diagonali, Boll. Un. Mat. Ital. Ser. Ⅳ, 6 (1972), 1–10. |
[67] | A. Porretta, Uniqueness and homogenization for a class of non coercive operators in divergence form, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 915–936. |
[68] | E. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, 1971. |
[69] | C. Trombetti, Existence and regularity for a class of non-uniformly elliptic equations in two dimensions, Differential Integral Equations, 13 (2000), 687–706. |
[70] | Z. Q. Yan, Everywhere regularity for solutions to quasilinear elliptic systems of triangular form, In: Partial differential equations, Berlin: Springer, 1988,255–261. https://doi.org/10.1007/BFb0082938 |
[71] | S. Zhou, A note on nonlinear elliptic systems involving measures, Electronic Journal of Differential Equations, 2000 (2000), 08. |
![]() |
Eutherian | Marsupial | |||
Mouse | Human | Opossum | ||
lncRNAs | Xist | √ | √ | × |
Ftx | √ | √ | × | |
Jpx | √ | √ | × | |
Tsix | √ | * | × | |
XACT | × | √ | × | |
Rsx | × | × | √ | |
√ = presence, × = absence, * = pseudogene. |